<description/> <author>Константин Титов</author> <company/> <keywords/> <revisedBy>Константин Титов</revisedBy> </userData> <identityInfo> <revision>8</revision> <documentID>40d2a37b-1824-4d9a-bfd0-9d0ca7bc3bf3</documentID> <versionID>0e78da8e-13eb-4e83-bbd9-2c8efd580d16</versionID> <parentVersionID>00000000-0000-0000-0000-000000000000</parentVersionID> <branchID>00000000-0000-0000-0000-000000000000</branchID> </identityInfo> </metadata> <settings> <presentation> <textRendering> <textStyles> <textStyle name="Normal"> <blockAttr margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit"/> <inlineAttr font-family="Arial" font-charset="0" font-size="10" font-weight="normal" font-style="normal" underline="false" line-through="false" vertical-align="baseline"/> </textStyle> <textStyle name="Heading 1"> <blockAttr margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit"/> 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<editor protection-level="none" view-annotations="false" view-regions="false"> <ruler is-visible="false" ruler-unit="cm"/> <plotTemplate> <xy item-idref="1"/> </plotTemplate> <grid granularity-x="6" granularity-y="6"/> </editor> <fileFormat image-type="image/png" image-quality="75" save-numeric-results="true" exclude-large-results="false" save-text-images="false" screen-dpi="96"/> <miscellaneous> <handbook handbook-region-tag-ub="281" can-delete-original-handbook-regions="true" can-delete-user-regions="true" can-print="true" can-copy="true" can-save="true" file-permission-mask="4294967295"/> </miscellaneous> </settings> <regions> <region region-id="281" left="36" top="24" width="469.5" height="526.5" align-x="36" align-y="24" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <ole clsid="{00020906-0000-0000-C000-000000000046}" type="embedded" item-idref="2"/> <rendering item-idref="3"/> </region> <region region-id="273" left="486" top="570.75" width="15.75" height="15.75" align-x="492.75" align-y="582" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:parens xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:real>1</ml:real> </ml:parens> </math> <rendering item-idref="4"/> </region> <region region-id="262" left="60" top="576.75" width="112.5" height="13.5" align-x="60" align-y="588" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Исследование ряда </f> </text> </region> <region region-id="263" left="198" top="573.75" width="245.25" height="50.25" align-x="243" align-y="600" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedBIUnit" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve">S</ml:id> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">N</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:plus/> <ml:real>1</ml:real> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:summation/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">n</ml:id> </ml:boundVars> <ml:parens> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:pow/> <ml:parens> <ml:real>-1</ml:real> </ml:parens> <ml:id xml:space="preserve">n</ml:id> </ml:apply> <ml:parens> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">n</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:parens> </ml:apply> <ml:parens> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">n</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:parens> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:apply> </ml:apply> </ml:parens> </ml:lambda> <ml:bounds> <ml:real>1</ml:real> <ml:id xml:space="preserve">N</ml:id> </ml:bounds> </ml:apply> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="5"/> </region> <region region-id="267" left="48" top="643.5" width="442.5" height="39.75" align-x="48" align-y="666" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">на сходимость проведем по критерию Коши:<sp count="2"/> </f> <region region-id="266" left="295.5" top="644.25" width="194.25" height="39" align-x="463.5" align-y="666" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:symEval style="default" hide-keywords="false" hide-lhs="false" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:limit/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">n</ml:id> </ml:boundVars> <ml:apply> <ml:nthRoot/> <ml:id xml:space="preserve">n</ml:id> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:mult style="default"/> <ml:parens> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">n</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:parens> <ml:parens> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">n</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:parens> </ml:apply> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:absval/> <ml:id xml:space="preserve">x</ml:id> </ml:apply> </ml:parens> <ml:id xml:space="preserve">n</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:lambda> <ml:id xml:space="preserve">∞</ml:id> </ml:apply> <ml:symResult> <ml:apply> <ml:absval/> <ml:id xml:space="preserve">x</ml:id> </ml:apply> </ml:symResult> </ml:symEval> </math> <rendering item-idref="6"/> </region> </text> </region> <region region-id="271" left="48" top="714" width="458.25" height="43.5" align-x="48" align-y="726" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">Таким образом, радиусом сходимости ряда (1) являются все значения </f> <region region-id="270" left="437.25" top="714.75" width="38.25" height="15.75" align-x="460.5" align-y="726" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:lessThan/> <ml:apply> <ml:absval/> <ml:id xml:space="preserve">x</ml:id> </ml:apply> <ml:real>1</ml:real> </ml:apply> </math> <rendering item-idref="7"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">. Для </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">ряда (1) нетрудно найти и его сумму при N->infinity, используя для этого </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">возможность символьных вычислений Mathcad </inlineAttr> </f> </text> </region> <region region-id="265" left="66" top="762" width="72" height="15.75" align-x="66" align-y="762" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <ole clsid="{0002CE03-0000-0000-C000-000000000046}" type="embedded" item-idref="8"/> <rendering item-idref="9"/> </region> <region region-id="280" left="468" top="792.75" width="15.75" height="15.75" align-x="474.75" align-y="804" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:parens xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:real>2</ml:real> </ml:parens> </math> <rendering item-idref="10"/> </region> <region region-id="274" left="60" top="783" width="292.5" height="51" align-x="300" align-y="810" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:symEval style="default" hide-keywords="false" hide-lhs="false" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:plus/> <ml:real>1</ml:real> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:summation/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">n</ml:id> </ml:boundVars> <ml:parens> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:pow/> <ml:parens> <ml:real>-1</ml:real> </ml:parens> <ml:id xml:space="preserve">n</ml:id> </ml:apply> <ml:parens> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">n</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:parens> </ml:apply> <ml:parens> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">n</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:parens> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:apply> </ml:apply> </ml:parens> </ml:lambda> <ml:bounds> <ml:real>1</ml:real> <ml:id xml:space="preserve">∞</ml:id> </ml:bounds> </ml:apply> </ml:apply> </ml:apply> <ml:command> <ml:id xml:space="preserve">simplify</ml:id> </ml:command> <ml:symResult> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">x</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:parens> <ml:real>3</ml:real> </ml:apply> </ml:apply> </ml:symResult> </ml:symEval> </math> <rendering item-idref="11"/> </region> <region region-id="275" left="60" top="849.75" width="390" height="39.75" align-x="60" align-y="870" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">Мы получили разложение функции </f> <region region-id="276" left="248.25" top="850.5" width="78.75" height="39" align-x="274.5" align-y="870" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve">f</ml:id> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">x</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:parens> <ml:real>3</ml:real> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="12"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> в биномиальный ряд.</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <sp/> </inlineAttr> </f> </text> </region> <region region-id="100" left="60" top="906.75" width="416.25" height="54" align-x="60" align-y="918" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr underline="false">Задача же имеет смысл только в том случае, когда неизвестна функция. </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr underline="false">Если </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr underline="false">же известна функция, то не нужен ряд! В этом смысле приводимая<sp count="2"/>за-</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr underline="false">дача искусственна: известен ряд и его сумма (функция). И тем не менее как</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr underline="false">пример она заслуживает внимание.</inlineAttr> </f> </text> </region> <region region-id="277" left="60" top="972.75" width="252.75" height="13.5" align-x="60" align-y="984" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Построим графики ряда S(x,N) и функции f(x) </f> </text> </region> <region region-id="207" left="48" top="1020" width="390" height="209.25" align-x="48" align-y="1020" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="13"/> <rendering item-idref="14"/> </region> <region region-id="279" left="36" top="1284.75" width="466.5" height="94.5" align-x="36" align-y="1296" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Совпадение графиков на рис. 1 хорошее, кроме границ.</f> <f family="Arial Cyr" charset="204" size="12">Проведём вычисление ряда S(x,N), приближающего функцию f(x) (предел, к которому стремится S(x,N) при </f> <f family="Arial Cyr" charset="204" size="12"> N -> infinity) с указанной точностью, и докажем его равномерную сходимость. Вообще, степенной ряд </f> <f family="Arial Cyr" charset="204" size="12">в радиусе его сходимости имеет абсолютную и равномерную сходимость по признаку Вейерштрасса (это доказывается). Для доказательства необходимо подобрать к </f> <f family="Arial Cyr" charset="204" size="12">этому ряду </f> <f family="Arial Cyr" charset="204" size="12">мажоранту, что для степенного ряда всегда возможно. </f> </text> </region> <region region-id="278" left="36" top="1422.75" width="447.75" height="67.5" align-x="36" align-y="1434" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Итак, раномерная сходимость есть. Остаётся оценить остаток ряда, который </f> <f family="Arial Cyr" charset="204" size="12">будет определять допускаемую погрешность при замене исходного ряда </f> <f family="Arial Cyr" charset="204" size="12">конечным числом его членов. Самое простое решение проводить оценку остатка </f> <f family="Arial Cyr" charset="204" size="12">ряда по остатку ряда мажоранты. Однако, это не всегда возможно, да и оценка</f> <f family="Arial Cyr" charset="204" size="12">получается грубой.</f> </text> </region> <region region-id="228" left="36" top="1506" width="470.25" height="78" align-x="36" align-y="1518" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Ряд S(x,N)<sp count="2"/> </f> <f family="Arial Cyr" charset="204" size="12">равномерно сходится<sp count="2"/>для всех </f> <region region-id="74" left="274.5" top="1506.75" width="38.25" height="15.75" align-x="297.75" align-y="1518" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:lessThan/> <ml:apply> <ml:absval/> <ml:id xml:space="preserve">x</ml:id> </ml:apply> <ml:real>1</ml:real> </ml:apply> </math> <rendering item-idref="15"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <sp/> </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> (</f> <f family="Arial Cyr" charset="204" size="12">сходиться к f(x) по определению</f> <f family="Arial Cyr" charset="204" size="12">).<sp count="2"/>Теперь, если задать точность </f> <f family="Arial Cyr" charset="204" size="12">приближения </f> <f family="Arial Cyr" charset="204" size="12">функции f(x)</f> <f family="Arial Cyr" charset="204" size="12"> рядом </f> <f family="Arial Cyr" charset="204" size="12">S(x,N) , например, как </f> <region region-id="75" left="36" top="1536.75" width="52.5" height="16.5" align-x="51" align-y="1548.75" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedBIFunction" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">ε</ml:id> <ml:real>0.001</ml:real> </ml:define> </math> <rendering item-idref="16"/> </region> <f family="Arial Cyr" charset="204" size="12">, то необходимо </f> <f family="Arial Cyr" charset="204" size="12">будет найти такое число N1>0, </f> <f family="Arial Cyr" charset="204" size="12">чтобы остаток ряда не превышал значения<sp count="2"/> </f> <region region-id="107" left="150.75" top="1554" width="7.5" height="16.5" align-x="153.75" align-y="1566" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:id xml:space="preserve" xmlns:ml="http://schemas.mathsoft.com/math20">ε</ml:id> </math> <rendering item-idref="17"/> </region> <f family="Arial Cyr" charset="204" size="12"> : </f> <region region-id="154" left="168" top="1554" width="126" height="16.5" align-x="278.25" align-y="1566" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:lessThan/> <ml:apply> <ml:absval/> <ml:apply> <ml:minus/> <ml:apply> <ml:id xml:space="preserve">f</ml:id> <ml:id xml:space="preserve">x</ml:id> </ml:apply> <ml:apply> <ml:id xml:space="preserve">S</ml:id> <ml:sequence> <ml:id xml:space="preserve">x</ml:id> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">N2</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:sequence> </ml:apply> </ml:apply> </ml:apply> <ml:id xml:space="preserve">ε</ml:id> </ml:apply> </math> <rendering item-idref="18"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">,<sp count="2"/>для </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">любого </inlineAttr> </f> <region region-id="229" left="369" top="1554.75" width="38.25" height="15.75" align-x="392.25" align-y="1566" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:lessThan/> <ml:apply> <ml:absval/> <ml:id xml:space="preserve">x</ml:id> </ml:apply> <ml:real>1</ml:real> </ml:apply> </math> <rendering item-idref="19"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> .<sp count="2"/>Г</inlineAttr> </f> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Arial Cyr" charset="204" size="12">де f(x) </f> </inlineAttr> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">обозначает предел S(x,N2->infiniti).<sp count="2"/> </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12">Запишем остаток </f> <f family="Arial Cyr" charset="204" size="12">ряда:</f> </text> </region> <region region-id="213" left="84" top="1611" width="233.25" height="51" align-x="132" align-y="1638" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve">r</ml:id> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">N1</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:summation/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">n</ml:id> </ml:boundVars> <ml:parens> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:pow/> <ml:parens> <ml:real>-1</ml:real> </ml:parens> <ml:id xml:space="preserve">n</ml:id> </ml:apply> <ml:parens> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">n</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:parens> </ml:apply> <ml:parens> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">n</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:parens> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:apply> </ml:apply> </ml:parens> </ml:lambda> <ml:bounds> <ml:id xml:space="preserve">N1</ml:id> <ml:id xml:space="preserve">∞</ml:id> </ml:bounds> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="20"/> </region> <region region-id="214" left="408" top="1626.75" width="15.75" height="15.75" align-x="414.75" align-y="1638" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:parens xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:real>3</ml:real> </ml:parens> </math> <rendering item-idref="21"/> </region> <region region-id="113" left="42" top="1674.75" width="457.5" height="54" align-x="42" align-y="1686" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">В данном случае оценка остатка ряда для 0<x<1 возможна, что не всегда выполнимо. </f> <f family="Arial Cyr" charset="204" size="12">Ряд (2) сходится, его члены монотонно убывают и имеют строгое чередование </f> <f family="Arial Cyr" charset="204" size="12">знака (при x>0). Поэтому остаток<sp count="2"/>ряда оценивается его первым </f> <f family="Arial Cyr" charset="204" size="12">членом, то есть </f> </text> </region> <region region-id="192" left="378" top="1758.75" width="21" height="15.75" align-x="387.75" align-y="1770" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:parens xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:mult style="auto-select"/> <ml:real>3</ml:real> <ml:id xml:space="preserve">a</ml:id> </ml:apply> </ml:parens> </math> <rendering item-idref="22"/> </region> <region region-id="115" left="66" top="1751.25" width="255" height="39" align-x="305.25" align-y="1776" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:lessThan/> <ml:apply> <ml:lessThan/> <ml:apply> <ml:absval/> <ml:apply> <ml:id xml:space="preserve">r</ml:id> <ml:sequence> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">N1</ml:id> </ml:sequence> </ml:apply> </ml:apply> <ml:apply> <ml:absval/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:apply> <ml:pow/> <ml:parens> <ml:real>-1</ml:real> </ml:parens> <ml:id xml:space="preserve">N1</ml:id> </ml:apply> <ml:real>2</ml:real> </ml:apply> <ml:parens> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">N1</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:parens> </ml:apply> <ml:parens> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">N1</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:parens> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">N1</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:id xml:space="preserve">ε</ml:id> </ml:apply> </math> <rendering item-idref="23"/> </region> <region region-id="230" left="54" top="1806.75" width="440.25" height="57.75" align-x="54" align-y="1818" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">Очевидно, что при заданном значении N1 наибольшему остатку<sp count="2"/>r(x,N1) будет соответствовать </f> <f family="Arial Cyr" charset="204" size="12">наибольшее значение х из интервала сходимости ряда, то есть</f> <f family="Arial Cyr" charset="204" size="12">x=sup(0,1). Следовательно, определяя N1<sp count="2"/>из условия<sp count="2"/>|r(sup(0,1),N1)| <</f> <region region-id="231" left="440.25" top="1834.5" width="7.5" height="16.5" align-x="443.25" align-y="1846.5" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:id xml:space="preserve" xmlns:ml="http://schemas.mathsoft.com/math20">ε</ml:id> </math> <rendering item-idref="24"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">, мы </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">заведомо удовлетворим требованию:</inlineAttr> </f> </text> </region> <region region-id="243" left="168" top="1884" width="68.25" height="15.75" align-x="168" align-y="1884" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <ole clsid="{0002CE03-0000-0000-C000-000000000046}" type="embedded" item-idref="25"/> <rendering item-idref="26"/> </region> <region region-id="242" left="258" top="1884" width="51" height="15.75" align-x="258" align-y="1884" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <ole clsid="{0002CE03-0000-0000-C000-000000000046}" type="embedded" item-idref="27"/> <rendering item-idref="28"/> </region> <region region-id="247" left="54" top="1920" width="440.25" height="30" align-x="54" align-y="1932" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Например, для любого </f> <region region-id="79" left="180.75" top="1920.75" width="58.5" height="15.75" align-x="215.25" align-y="1932" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:lessThan/> <ml:apply> <ml:lessThan/> <ml:real>0</ml:real> <ml:id xml:space="preserve">x</ml:id> </ml:apply> <ml:real>0.9</ml:real> </ml:apply> </math> <rendering item-idref="29"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <sp count="2"/> </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">находим N1 </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12">обычным подбором</f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <sp/> </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> или </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">по алгоритму:</inlineAttr> </f> </text> </region> <region region-id="217" left="72" top="1980.75" width="204.75" height="96.75" align-x="95.25" align-y="1992" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">N1</ml:id> <ml:program> <ml:localDefine> <ml:id xml:space="preserve">n</ml:id> <ml:real>1</ml:real> </ml:localDefine> <ml:while> <ml:apply> <ml:greaterThan/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>2</ml:real> </ml:apply> <ml:parens> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">n</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:parens> </ml:apply> <ml:parens> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">n</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:parens> </ml:apply> <ml:apply> <ml:pow/> <ml:real>0.9</ml:real> <ml:id xml:space="preserve">n</ml:id> </ml:apply> </ml:apply> <ml:id xml:space="preserve">ε</ml:id> </ml:apply> <ml:localDefine> <ml:id xml:space="preserve">n</ml:id> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">n</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:localDefine> </ml:while> <ml:id xml:space="preserve">n</ml:id> </ml:program> </ml:define> </math> <rendering item-idref="30"/> </region> <region region-id="253" left="42" top="2112" width="448.5" height="30" align-x="42" align-y="2124" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Получили: </f> <region region-id="223" left="98.25" top="2112.75" width="56.25" height="15.75" align-x="120.75" align-y="2124" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">N1</ml:id> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>155</ml:real> </result> </ml:eval> </math> <rendering item-idref="31"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">. </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12">Введём обозначение первого члена ряда (3), как его оценку:</f> </text> </region> <region region-id="155" left="54" top="2152.5" width="208.5" height="33.75" align-x="105" align-y="2172" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedBIUnit" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve">T</ml:id> <ml:boundVars> <ml:id xml:space="preserve">N1</ml:id> <ml:id xml:space="preserve">x</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>2</ml:real> </ml:apply> <ml:parens> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">N1</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:parens> </ml:apply> <ml:parens> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">N1</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:parens> </ml:apply> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:absval/> <ml:id xml:space="preserve">x</ml:id> </ml:apply> </ml:parens> <ml:id xml:space="preserve">N1</ml:id> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="32"/> </region> <region region-id="193" left="390" top="2160.75" width="15.75" height="15.75" align-x="396.75" align-y="2172" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:parens xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:real>4</ml:real> </ml:parens> </math> <rendering item-idref="33"/> </region> <region region-id="248" left="42" top="2190.75" width="460.5" height="35.25" align-x="42" align-y="2208" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">Оценка остатка ряда (3) для х=0.9 равна: </f> <region region-id="250" left="270.75" top="2191.5" width="143.25" height="21" align-x="333.75" align-y="2208" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:id xml:space="preserve">T</ml:id> <ml:sequence> <ml:real>155</ml:real> <ml:real>0.9</ml:real> </ml:sequence> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>0.00098988152380112332</ml:real> </result> </ml:eval> </math> <rendering item-idref="34"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">. Для остальных</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">же значений 0<x<0.9 </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> о</f> <f family="Arial Cyr" charset="204" size="12">ценка остатка ряда (3) будет ещё меньше.</f> </text> </region> <region region-id="220" left="42" top="2238" width="456" height="43.5" align-x="42" align-y="2250" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Выражая N1 через системную переменную FRAME как </f> <region region-id="221" left="346.5" top="2238.75" width="114.75" height="15.75" align-x="369.75" align-y="2250" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedUDScalar" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">N1</ml:id> <ml:apply> <ml:plus/> <ml:real>155</ml:real> <ml:apply> <ml:mult style="default"/> <ml:real>2</ml:real> <ml:id xml:space="preserve">FRAME</ml:id> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="35"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> ,</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">можно также отснять видеоклип изменения в интерактивном режиме графиков</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">на рис. 2 - 4.</inlineAttr> </f> </text> </region> <region region-id="209" left="30" top="2297.25" width="479.25" height="119.25" align-x="30" align-y="2310" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Итак, остаток ряда (3) не превышает значения </f> <region region-id="189" left="288" top="2298" width="7.5" height="16.5" align-x="291" align-y="2310" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:id xml:space="preserve" xmlns:ml="http://schemas.mathsoft.com/math20">ε</ml:id> </math> <rendering item-idref="36"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">. Он сходится абсолютно и </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">раномерно, так как мажорируем абсолютно сходящимся числовым<sp count="2"/>рядом с положительными членами, например, такими:<sp count="2"/> </inlineAttr> </f> <region region-id="191" left="283.5" top="2328.75" width="117.75" height="33.75" align-x="369" align-y="2348.25" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:mult style="default"/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>2</ml:real> </ml:apply> <ml:parens> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">n</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:parens> </ml:apply> <ml:parens> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">n</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:parens> </ml:apply> <ml:apply> <ml:pow/> <ml:parens> <ml:real>0.9</ml:real> </ml:parens> <ml:id xml:space="preserve">n</ml:id> </ml:apply> </ml:apply> </math> <rendering item-idref="37"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">.<sp count="2"/>Это означает, </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">что остаток </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">ряда </inlineAttr> </f> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Arial Cyr" charset="204" size="12">(3) с любыми -0.9<x<0.9 также будет абсолютно и равномерно сходиться. </f> </inlineAttr> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">Причём в этом случае для оценки остатка ряда (3) не обязательно как</inlineAttr> </f> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Arial Cyr" charset="204" size="12">было указано выше брать остаток мажоранты, который иногда трудно оценить, а можно использовать формулу (3a), что и было сделано.</f> </inlineAttr> </text> </region> <region region-id="255" left="30" top="2436.75" width="440.25" height="70.5" align-x="30" align-y="2448" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Таким образом, беря в ряду S(x,N)<sp count="2"/>N=154 его членов и вычисляя значение</f> <f family="Arial Cyr" charset="204" size="12">такого ряда, мы получим приближенное значение функции f(x) с точностью не</f> <f family="Arial Cyr" charset="204" size="12">хуже 0.001 для любых </f> <region region-id="88" left="152.25" top="2464.5" width="58.5" height="15.75" align-x="186.75" align-y="2475.75" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:lessThan/> <ml:apply> <ml:lessThan/> <ml:real>0</ml:real> <ml:id xml:space="preserve">x</ml:id> </ml:apply> <ml:real>0.9</ml:real> </ml:apply> </math> <rendering item-idref="38"/> </region> <f family="Arial Cyr" charset="204" size="12">. Графическая иллюстрация полученного</f> <f family="Arial Cyr" charset="204" size="12">результата дана на рис.2.<sp count="2"/>Для<sp count="2"/>-1<x<0 вопрос оценки остатка ряда остается открытым!</f> </text> </region> <region region-id="194" left="42" top="2526" width="433.5" height="149.25" align-x="42" align-y="2526" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="39"/> <rendering item-idref="40"/> </region> <region region-id="138" left="42" top="2712.75" width="476.25" height="27" align-x="42" align-y="2724" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Полученное приближение намного точнее расчётного (тем более для </f> <f family="Arial Cyr" charset="204" size="12">мажоранты). Его </f> <f family="Arial Cyr" charset="204" size="12">следует показать отдельно на графике рис.3 и 4.</f> </text> </region> <region region-id="148" left="42" top="2766.75" width="123" height="15.75" align-x="87" align-y="2778" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve">E</ml:id> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">N</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:minus/> <ml:apply> <ml:id xml:space="preserve">f</ml:id> <ml:id xml:space="preserve">x</ml:id> </ml:apply> <ml:apply> <ml:id xml:space="preserve">S</ml:id> <ml:sequence> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">N</ml:id> </ml:sequence> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="41"/> </region> <region region-id="157" left="192" top="2766.75" width="112.5" height="15.75" align-x="261.75" align-y="2778" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:id xml:space="preserve">E</ml:id> <ml:sequence> <ml:real>-0.9</ml:real> <ml:real>154</ml:real> </ml:sequence> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>0.011106460996757051</ml:real> </result> </ml:eval> </math> <rendering item-idref="42"/> </region> <region region-id="196" left="324" top="2761.5" width="150" height="21" align-x="387" align-y="2778" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:id xml:space="preserve">E</ml:id> <ml:sequence> <ml:real>0.9</ml:real> <ml:real>154</ml:real> </ml:sequence> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>-0.00051783591292875464</ml:real> </result> </ml:eval> </math> <rendering item-idref="43"/> </region> <region region-id="195" left="36" top="2820" width="432.75" height="278.25" align-x="36" align-y="2820" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="44"/> <rendering item-idref="45"/> </region> <region region-id="256" left="48" top="3144" width="420.75" height="248.25" align-x="48" align-y="3144" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="46"/> <rendering item-idref="47"/> </region> <region region-id="199" left="66" top="3426.75" width="420" height="61.5" align-x="66" align-y="3438" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Графики E(x,N) показывают, что точность приближения сильно зависит от </f> <f family="Arial Cyr" charset="204" size="12">х. Но всегда можно выбрать такое значение </f> <region region-id="200" left="311.25" top="3441" width="27.75" height="16.5" align-x="320.25" align-y="3453" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">N</ml:id> <ml:id xml:space="preserve">ε</ml:id> </ml:apply> </math> <rendering item-idref="48"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">, что при n>N </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">будет выполняться </inlineAttr> </f> <region region-id="201" left="174.75" top="3458.25" width="101.25" height="16.5" align-x="260.25" align-y="3470.25" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:lessThan/> <ml:apply> <ml:absval/> <ml:apply> <ml:minus/> <ml:apply> <ml:id xml:space="preserve">f</ml:id> <ml:id xml:space="preserve">x</ml:id> </ml:apply> <ml:apply> <ml:id xml:space="preserve">S</ml:id> <ml:sequence> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">N</ml:id> </ml:sequence> </ml:apply> </ml:apply> </ml:apply> <ml:id xml:space="preserve">ε</ml:id> </ml:apply> </math> <rendering item-idref="49"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <sp/> </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> для всех </f> <region region-id="202" left="333" top="3459" width="38.25" height="15.75" align-x="356.25" align-y="3470.25" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:lessThan/> <ml:apply> <ml:absval/> <ml:id xml:space="preserve">x</ml:id> </ml:apply> <ml:real>1</ml:real> </ml:apply> </math> <rendering item-idref="50"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">. Это свойство равномерной сходимости<sp count="2"/>также следует из графиков рис. 3 и 4.</inlineAttr> </f> </text> </region> <region region-id="177" left="78" top="3516.75" width="219" height="13.5" align-x="78" align-y="3528" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">Ниже приводятся данные для сранения:</f> </text> </region> <region region-id="176" left="78" top="3553.5" width="156" height="21" align-x="153.75" align-y="3570" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:id xml:space="preserve">E</ml:id> <ml:sequence> <ml:real>-0.85</ml:real> <ml:real>154</ml:real> </ml:sequence> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>1.0077544629893964E-06</ml:real> </result> </ml:eval> </math> <rendering item-idref="51"/> </region> <region region-id="173" left="288" top="3553.5" width="156" height="21" align-x="357" align-y="3570" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:id xml:space="preserve">E</ml:id> <ml:sequence> <ml:real>0.85</ml:real> <ml:real>154</ml:real> </ml:sequence> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>-7.5543619304507814E-08</ml:real> </result> </ml:eval> </math> <rendering item-idref="52"/> </region> <region region-id="175" left="78" top="3595.5" width="155.25" height="21" align-x="147.75" align-y="3612" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:id xml:space="preserve">E</ml:id> <ml:sequence> <ml:real>-0.8</ml:real> <ml:real>154</ml:real> </ml:sequence> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>6.127720553195104E-11</ml:real> </result> </ml:eval> </math> <rendering item-idref="53"/> </region> <region region-id="172" left="288" top="3595.5" width="155.25" height="21" align-x="351" align-y="3612" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:id xml:space="preserve">E</ml:id> <ml:sequence> <ml:real>0.8</ml:real> <ml:real>154</ml:real> </ml:sequence> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>-6.4458993698224276E-12</ml:real> </result> </ml:eval> </math> <rendering item-idref="54"/> </region> <region region-id="174" left="78" top="3631.5" width="155.25" height="21" align-x="147.75" align-y="3648" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:id xml:space="preserve">E</ml:id> <ml:sequence> <ml:real>-0.7</ml:real> <ml:real>154</ml:real> </ml:sequence> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>2.8421709430404007E-14</ml:real> </result> </ml:eval> </math> <rendering item-idref="55"/> </region> <region region-id="171" left="288" top="3636.75" width="84.75" height="15.75" align-x="351" align-y="3648" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:id xml:space="preserve">E</ml:id> <ml:sequence> <ml:real>0.7</ml:real> <ml:real>154</ml:real> </ml:sequence> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>-8.3266726846886741E-17</ml:real> </result> </ml:eval> </math> <rendering item-idref="56"/> </region> <region region-id="187" left="66" top="3672" width="221.25" height="16.5" align-x="66" align-y="3684" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Для значений </f> <region region-id="181" left="141.75" top="3672.75" width="69" height="15.75" align-x="195.75" align-y="3684" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:lessThan/> <ml:apply> <ml:lessThan/> <ml:real>0.9</ml:real> <ml:apply> <ml:absval/> <ml:id xml:space="preserve">x</ml:id> </ml:apply> </ml:apply> <ml:real>1</ml:real> </ml:apply> </math> <rendering item-idref="57"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> будем иметь:</inlineAttr> </f> </text> </region> <region region-id="183" left="78" top="3702.75" width="130.5" height="15.75" align-x="153.75" align-y="3714" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:id xml:space="preserve">E</ml:id> <ml:sequence> <ml:real>-0.95</ml:real> <ml:real>154</ml:real> </ml:sequence> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>109.90942368342712</ml:real> </result> </ml:eval> </math> <rendering item-idref="58"/> </region> <region region-id="184" left="288" top="3702.75" width="106.5" height="15.75" align-x="357" align-y="3714" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:id xml:space="preserve">E</ml:id> <ml:sequence> <ml:real>0.95</ml:real> <ml:real>154</ml:real> </ml:sequence> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>-2.1999148483361424</ml:real> </result> </ml:eval> </math> <rendering item-idref="59"/> </region> <region region-id="186" left="60" top="3738.75" width="440.25" height="27" align-x="60" align-y="3750" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Следовательно, в последнем случае количество членов ряда<sp count="2"/>S(x,N) надо брать 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item-idref="3"/> </region> <region region-id="273" left="486" top="570.75" width="15.75" height="15.75" align-x="492.75" align-y="582" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:parens xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:real>1</ml:real> </ml:parens> </math> <rendering item-idref="4"/> </region> <region region-id="262" left="60" top="576.75" width="112.5" height="13.5" align-x="60" align-y="588" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Исследование ряда </f> </text> </region> <region region-id="263" left="198" top="573.75" width="245.25" height="50.25" align-x="243" align-y="600" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedBIUnit" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve">S</ml:id> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">N</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:plus/> <ml:real>1</ml:real> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:summation/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">n</ml:id> </ml:boundVars> <ml:parens> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:pow/> <ml:parens> <ml:real>-1</ml:real> </ml:parens> <ml:id xml:space="preserve">n</ml:id> </ml:apply> <ml:parens> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">n</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:parens> </ml:apply> <ml:parens> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">n</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:parens> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:apply> </ml:apply> </ml:parens> </ml:lambda> <ml:bounds> <ml:real>1</ml:real> <ml:id xml:space="preserve">N</ml:id> </ml:bounds> </ml:apply> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="5"/> </region> <region region-id="267" left="48" top="643.5" width="442.5" height="39.75" align-x="48" align-y="666" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">на сходимость проведем по критерию Коши:<sp count="2"/> </f> <region region-id="266" left="295.5" top="644.25" width="194.25" height="39" align-x="463.5" align-y="666" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:symEval style="default" hide-keywords="false" hide-lhs="false" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:limit/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">n</ml:id> </ml:boundVars> <ml:apply> <ml:nthRoot/> <ml:id xml:space="preserve">n</ml:id> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:mult style="default"/> <ml:parens> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">n</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:parens> <ml:parens> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">n</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:parens> </ml:apply> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:absval/> <ml:id xml:space="preserve">x</ml:id> </ml:apply> </ml:parens> <ml:id xml:space="preserve">n</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:lambda> <ml:id xml:space="preserve">∞</ml:id> </ml:apply> <ml:symResult> <ml:apply> <ml:absval/> <ml:id xml:space="preserve">x</ml:id> </ml:apply> </ml:symResult> </ml:symEval> </math> <rendering item-idref="6"/> </region> </text> </region> <region region-id="271" left="48" top="714" width="458.25" height="43.5" align-x="48" align-y="726" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">Таким образом, радиусом сходимости ряда (1) являются все значения </f> <region region-id="270" left="437.25" top="714.75" width="38.25" height="15.75" align-x="460.5" align-y="726" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:lessThan/> <ml:apply> <ml:absval/> <ml:id xml:space="preserve">x</ml:id> </ml:apply> <ml:real>1</ml:real> </ml:apply> </math> <rendering item-idref="7"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">. Для </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">ряда (1) нетрудно найти и его сумму при N->infinity, используя для этого </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">возможность символьных вычислений Mathcad </inlineAttr> </f> </text> </region> <region region-id="265" left="66" top="762" width="72" height="15.75" align-x="66" align-y="762" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <ole clsid="{0002CE03-0000-0000-C000-000000000046}" type="embedded" item-idref="8"/> <rendering item-idref="9"/> </region> <region region-id="280" left="468" top="792.75" width="15.75" height="15.75" align-x="474.75" align-y="804" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:parens xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:real>2</ml:real> </ml:parens> </math> <rendering item-idref="10"/> </region> <region region-id="274" left="60" top="783" width="292.5" height="51" align-x="300" align-y="810" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:symEval style="default" hide-keywords="false" hide-lhs="false" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:plus/> <ml:real>1</ml:real> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:summation/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">n</ml:id> </ml:boundVars> <ml:parens> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:pow/> <ml:parens> <ml:real>-1</ml:real> </ml:parens> <ml:id xml:space="preserve">n</ml:id> </ml:apply> <ml:parens> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">n</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:parens> </ml:apply> <ml:parens> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">n</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:parens> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:apply> </ml:apply> </ml:parens> </ml:lambda> <ml:bounds> <ml:real>1</ml:real> <ml:id xml:space="preserve">∞</ml:id> </ml:bounds> </ml:apply> </ml:apply> </ml:apply> <ml:command> <ml:id xml:space="preserve">simplify</ml:id> </ml:command> <ml:symResult> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">x</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:parens> <ml:real>3</ml:real> </ml:apply> </ml:apply> </ml:symResult> </ml:symEval> </math> <rendering item-idref="11"/> </region> <region region-id="275" left="60" top="849.75" width="390" height="39.75" align-x="60" align-y="870" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">Мы получили разложение функции </f> <region region-id="276" left="248.25" top="850.5" width="78.75" height="39" align-x="274.5" align-y="870" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve">f</ml:id> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">x</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:parens> <ml:real>3</ml:real> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="12"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> в биномиальный ряд.</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <sp/> </inlineAttr> </f> </text> </region> <region region-id="100" left="60" top="906.75" width="416.25" height="54" align-x="60" align-y="918" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr underline="false">Задача же имеет смысл только в том случае, когда неизвестна функция. </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr underline="false">Если </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr underline="false">же известна функция, то не нужен ряд! В этом смысле приводимая<sp count="2"/>за-</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr underline="false">дача искусственна: известен ряд и его сумма (функция). И тем не менее как</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr underline="false">пример она заслуживает внимание.</inlineAttr> </f> </text> </region> <region region-id="277" left="60" top="972.75" width="252.75" height="13.5" align-x="60" align-y="984" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Построим графики ряда S(x,N) и функции f(x) </f> </text> </region> <region region-id="207" left="48" top="1020" width="390" height="209.25" align-x="48" align-y="1020" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="13"/> <rendering item-idref="14"/> </region> <region region-id="279" left="36" top="1284.75" width="466.5" height="94.5" align-x="36" align-y="1296" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Совпадение графиков на рис. 1 хорошее, кроме границ.</f> <f family="Arial Cyr" charset="204" size="12">Проведём вычисление ряда S(x,N), приближающего функцию f(x) (предел, к которому стремится S(x,N) при </f> <f family="Arial Cyr" charset="204" size="12"> N -> infinity) с указанной точностью, и докажем его равномерную сходимость. Вообще, степенной ряд </f> <f family="Arial Cyr" charset="204" size="12">в радиусе его сходимости имеет абсолютную и равномерную сходимость по признаку Вейерштрасса (это доказывается). Для доказательства необходимо подобрать к </f> <f family="Arial Cyr" charset="204" size="12">этому ряду </f> <f family="Arial Cyr" charset="204" size="12">мажоранту, что для степенного ряда всегда возможно. </f> </text> </region> <region region-id="278" left="36" top="1422.75" width="447.75" height="67.5" align-x="36" align-y="1434" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Итак, раномерная сходимость есть. Остаётся оценить остаток ряда, который </f> <f family="Arial Cyr" charset="204" size="12">будет определять допускаемую погрешность при замене исходного ряда </f> <f family="Arial Cyr" charset="204" size="12">конечным числом его членов. Самое простое решение проводить оценку остатка </f> <f family="Arial Cyr" charset="204" size="12">ряда по остатку ряда мажоранты. Однако, это не всегда возможно, да и оценка</f> <f family="Arial Cyr" charset="204" size="12">получается грубой.</f> </text> </region> <region region-id="228" left="36" top="1506" width="470.25" height="78" align-x="36" align-y="1518" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Ряд S(x,N)<sp count="2"/> </f> <f family="Arial Cyr" charset="204" size="12">равномерно сходится<sp count="2"/>для всех </f> <region region-id="74" left="274.5" top="1506.75" width="38.25" height="15.75" align-x="297.75" align-y="1518" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:lessThan/> <ml:apply> <ml:absval/> <ml:id xml:space="preserve">x</ml:id> </ml:apply> <ml:real>1</ml:real> </ml:apply> </math> <rendering item-idref="15"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <sp/> </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> (</f> <f family="Arial Cyr" charset="204" size="12">сходиться к f(x) по определению</f> <f family="Arial Cyr" charset="204" size="12">).<sp count="2"/>Теперь, если задать точность </f> <f family="Arial Cyr" charset="204" size="12">приближения </f> <f family="Arial Cyr" charset="204" size="12">функции f(x)</f> <f family="Arial Cyr" charset="204" size="12"> рядом </f> <f family="Arial Cyr" charset="204" size="12">S(x,N) , например, как </f> <region region-id="75" left="36" top="1536.75" width="52.5" height="16.5" align-x="51" align-y="1548.75" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedBIFunction" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">ε</ml:id> <ml:real>0.001</ml:real> </ml:define> </math> <rendering item-idref="16"/> </region> <f family="Arial Cyr" charset="204" size="12">, то необходимо </f> <f family="Arial Cyr" charset="204" size="12">будет найти такое число N1>0, </f> <f family="Arial Cyr" charset="204" size="12">чтобы остаток ряда не превышал значения<sp count="2"/> </f> <region region-id="107" left="150.75" top="1554" width="7.5" height="16.5" align-x="153.75" align-y="1566" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:id xml:space="preserve" xmlns:ml="http://schemas.mathsoft.com/math20">ε</ml:id> </math> <rendering item-idref="17"/> </region> <f family="Arial Cyr" charset="204" size="12"> : </f> <region region-id="154" left="168" top="1554" width="126" height="16.5" align-x="278.25" align-y="1566" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:lessThan/> <ml:apply> <ml:absval/> <ml:apply> <ml:minus/> <ml:apply> <ml:id xml:space="preserve">f</ml:id> <ml:id xml:space="preserve">x</ml:id> </ml:apply> <ml:apply> <ml:id xml:space="preserve">S</ml:id> <ml:sequence> <ml:id xml:space="preserve">x</ml:id> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">N2</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:sequence> </ml:apply> </ml:apply> </ml:apply> <ml:id xml:space="preserve">ε</ml:id> </ml:apply> </math> <rendering item-idref="18"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">,<sp count="2"/>для </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">любого </inlineAttr> </f> <region region-id="229" left="369" top="1554.75" width="38.25" height="15.75" align-x="392.25" align-y="1566" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:lessThan/> <ml:apply> <ml:absval/> <ml:id xml:space="preserve">x</ml:id> </ml:apply> <ml:real>1</ml:real> </ml:apply> </math> <rendering item-idref="19"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> .<sp count="2"/>Г</inlineAttr> </f> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Arial Cyr" charset="204" size="12">де f(x) </f> </inlineAttr> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">обозначает предел S(x,N2->infiniti).<sp count="2"/> </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12">Запишем остаток </f> <f family="Arial Cyr" charset="204" size="12">ряда:</f> </text> </region> <region region-id="213" left="84" top="1611" width="233.25" height="51" align-x="132" align-y="1638" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve">r</ml:id> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">N1</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:summation/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">n</ml:id> </ml:boundVars> <ml:parens> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:pow/> <ml:parens> <ml:real>-1</ml:real> </ml:parens> <ml:id xml:space="preserve">n</ml:id> </ml:apply> <ml:parens> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">n</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:parens> </ml:apply> <ml:parens> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">n</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:parens> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:apply> </ml:apply> </ml:parens> </ml:lambda> <ml:bounds> <ml:id xml:space="preserve">N1</ml:id> <ml:id xml:space="preserve">∞</ml:id> </ml:bounds> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="20"/> </region> <region region-id="214" left="408" top="1626.75" width="15.75" height="15.75" align-x="414.75" align-y="1638" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:parens xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:real>3</ml:real> </ml:parens> </math> <rendering item-idref="21"/> </region> <region region-id="113" left="42" top="1674.75" width="457.5" height="54" align-x="42" align-y="1686" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">В данном случае оценка остатка ряда для 0<x<1 возможна, что не всегда выполнимо. </f> <f family="Arial Cyr" charset="204" size="12">Ряд (2) сходится, его члены монотонно убывают и имеют строгое чередование </f> <f family="Arial Cyr" charset="204" size="12">знака (при x>0). Поэтому остаток<sp count="2"/>ряда оценивается его первым </f> <f family="Arial Cyr" charset="204" size="12">членом, то есть </f> </text> </region> <region region-id="192" left="378" top="1758.75" width="21" height="15.75" align-x="387.75" align-y="1770" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:parens xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:mult style="auto-select"/> <ml:real>3</ml:real> <ml:id xml:space="preserve">a</ml:id> </ml:apply> </ml:parens> </math> <rendering item-idref="22"/> </region> <region region-id="115" left="66" top="1751.25" width="255" height="39" align-x="305.25" align-y="1776" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:lessThan/> <ml:apply> <ml:lessThan/> <ml:apply> <ml:absval/> <ml:apply> <ml:id xml:space="preserve">r</ml:id> <ml:sequence> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">N1</ml:id> </ml:sequence> </ml:apply> </ml:apply> <ml:apply> <ml:absval/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:apply> <ml:pow/> <ml:parens> <ml:real>-1</ml:real> </ml:parens> <ml:id xml:space="preserve">N1</ml:id> </ml:apply> <ml:real>2</ml:real> </ml:apply> <ml:parens> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">N1</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:parens> </ml:apply> <ml:parens> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">N1</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:parens> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">N1</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:id xml:space="preserve">ε</ml:id> </ml:apply> </math> <rendering item-idref="23"/> </region> <region region-id="230" left="54" top="1806.75" width="440.25" height="57.75" align-x="54" align-y="1818" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">Очевидно, что при заданном значении N1 наибольшему остатку<sp count="2"/>r(x,N1) будет соответствовать </f> <f family="Arial Cyr" charset="204" size="12">наибольшее значение х из интервала сходимости ряда, то есть</f> <f family="Arial Cyr" charset="204" size="12">x=sup(0,1). Следовательно, определяя N1<sp count="2"/>из условия<sp count="2"/>|r(sup(0,1),N1)| <</f> <region region-id="231" left="440.25" top="1834.5" width="7.5" height="16.5" align-x="443.25" align-y="1846.5" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:id xml:space="preserve" xmlns:ml="http://schemas.mathsoft.com/math20">ε</ml:id> </math> <rendering item-idref="24"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">, мы </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">заведомо удовлетворим требованию:</inlineAttr> </f> </text> </region> <region region-id="243" left="168" top="1884" width="68.25" height="15.75" align-x="168" align-y="1884" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <ole clsid="{0002CE03-0000-0000-C000-000000000046}" type="embedded" item-idref="25"/> <rendering item-idref="26"/> </region> <region region-id="242" left="258" top="1884" width="51" height="15.75" align-x="258" align-y="1884" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <ole clsid="{0002CE03-0000-0000-C000-000000000046}" type="embedded" item-idref="27"/> <rendering item-idref="28"/> </region> <region region-id="247" left="54" top="1920" width="440.25" height="30" align-x="54" align-y="1932" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Например, для любого </f> <region region-id="79" left="180.75" top="1920.75" width="58.5" height="15.75" align-x="215.25" align-y="1932" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:lessThan/> <ml:apply> <ml:lessThan/> <ml:real>0</ml:real> <ml:id xml:space="preserve">x</ml:id> </ml:apply> <ml:real>0.9</ml:real> </ml:apply> </math> <rendering item-idref="29"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <sp count="2"/> </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">находим N1 </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12">обычным подбором</f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <sp/> </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> или </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">по алгоритму:</inlineAttr> </f> </text> </region> <region region-id="217" left="72" top="1980.75" width="204.75" height="96.75" align-x="95.25" align-y="1992" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">N1</ml:id> <ml:program> <ml:localDefine> <ml:id xml:space="preserve">n</ml:id> <ml:real>1</ml:real> </ml:localDefine> <ml:while> <ml:apply> <ml:greaterThan/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>2</ml:real> </ml:apply> <ml:parens> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">n</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:parens> </ml:apply> <ml:parens> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">n</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:parens> </ml:apply> <ml:apply> <ml:pow/> <ml:real>0.9</ml:real> <ml:id xml:space="preserve">n</ml:id> </ml:apply> </ml:apply> <ml:id xml:space="preserve">ε</ml:id> </ml:apply> <ml:localDefine> <ml:id xml:space="preserve">n</ml:id> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">n</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:localDefine> </ml:while> <ml:id xml:space="preserve">n</ml:id> </ml:program> </ml:define> </math> <rendering item-idref="30"/> </region> <region region-id="253" left="42" top="2112" width="448.5" height="30" align-x="42" align-y="2124" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Получили: </f> <region region-id="223" left="98.25" top="2112.75" width="56.25" height="15.75" align-x="120.75" align-y="2124" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">N1</ml:id> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>155</ml:real> </result> </ml:eval> </math> <rendering item-idref="31"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">. </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12">Введём обозначение первого члена ряда (3), как его оценку:</f> </text> </region> <region region-id="155" left="54" top="2152.5" width="208.5" height="33.75" align-x="105" align-y="2172" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedBIUnit" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve">T</ml:id> <ml:boundVars> <ml:id xml:space="preserve">N1</ml:id> <ml:id xml:space="preserve">x</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>2</ml:real> </ml:apply> <ml:parens> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">N1</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:parens> </ml:apply> <ml:parens> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">N1</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:parens> </ml:apply> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:absval/> <ml:id xml:space="preserve">x</ml:id> </ml:apply> </ml:parens> <ml:id xml:space="preserve">N1</ml:id> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="32"/> </region> <region region-id="193" left="390" top="2160.75" width="15.75" height="15.75" align-x="396.75" align-y="2172" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:parens xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:real>4</ml:real> </ml:parens> </math> <rendering item-idref="33"/> </region> <region region-id="248" left="42" top="2190.75" width="460.5" height="35.25" align-x="42" align-y="2208" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">Оценка остатка ряда (3) для х=0.9 равна: </f> <region region-id="250" left="270.75" top="2191.5" width="143.25" height="21" align-x="333.75" align-y="2208" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:id xml:space="preserve">T</ml:id> <ml:sequence> <ml:real>155</ml:real> <ml:real>0.9</ml:real> </ml:sequence> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>0.00098988152380112332</ml:real> </result> </ml:eval> </math> <rendering item-idref="34"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">. Для остальных</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">же значений 0<x<0.9 </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> о</f> <f family="Arial Cyr" charset="204" size="12">ценка остатка ряда (3) будет ещё меньше.</f> </text> </region> <region region-id="220" left="42" top="2238" width="456" height="43.5" align-x="42" align-y="2250" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Выражая N1 через системную переменную FRAME как </f> <region region-id="221" left="346.5" top="2238.75" width="114.75" height="15.75" align-x="369.75" align-y="2250" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedUDScalar" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">N1</ml:id> <ml:apply> <ml:plus/> <ml:real>155</ml:real> <ml:apply> <ml:mult style="default"/> <ml:real>2</ml:real> <ml:id xml:space="preserve">FRAME</ml:id> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="35"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> ,</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">можно также отснять видеоклип изменения в интерактивном режиме графиков</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">на рис. 2 - 4.</inlineAttr> </f> </text> </region> <region region-id="209" left="30" top="2297.25" width="479.25" height="119.25" align-x="30" align-y="2310" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Итак, остаток ряда (3) не превышает значения </f> <region region-id="189" left="288" top="2298" width="7.5" height="16.5" align-x="291" align-y="2310" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:id xml:space="preserve" xmlns:ml="http://schemas.mathsoft.com/math20">ε</ml:id> </math> <rendering item-idref="36"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">. Он сходится абсолютно и </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">раномерно, так как мажорируем абсолютно сходящимся числовым<sp count="2"/>рядом с положительными членами, например, такими:<sp count="2"/> </inlineAttr> </f> <region region-id="191" left="283.5" top="2328.75" width="117.75" height="33.75" align-x="369" align-y="2348.25" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:mult style="default"/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>2</ml:real> </ml:apply> <ml:parens> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">n</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:parens> </ml:apply> <ml:parens> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">n</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:parens> </ml:apply> <ml:apply> <ml:pow/> <ml:parens> <ml:real>0.9</ml:real> </ml:parens> <ml:id xml:space="preserve">n</ml:id> </ml:apply> </ml:apply> </math> <rendering item-idref="37"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">.<sp count="2"/>Это означает, </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">что остаток </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">ряда </inlineAttr> </f> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Arial Cyr" charset="204" size="12">(3) с любыми -0.9<x<0.9 также будет абсолютно и равномерно сходиться. </f> </inlineAttr> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">Причём в этом случае для оценки остатка ряда (3) не обязательно как</inlineAttr> </f> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Arial Cyr" charset="204" size="12">было указано выше брать остаток мажоранты, который иногда трудно оценить, а можно использовать формулу (3a), что и было сделано.</f> </inlineAttr> </text> </region> <region region-id="255" left="30" top="2436.75" width="440.25" height="70.5" align-x="30" align-y="2448" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Таким образом, беря в ряду S(x,N)<sp count="2"/>N=154 его членов и вычисляя значение</f> <f family="Arial Cyr" charset="204" size="12">такого ряда, мы получим приближенное значение функции f(x) с точностью не</f> <f family="Arial Cyr" charset="204" size="12">хуже 0.001 для любых </f> <region region-id="88" left="152.25" top="2464.5" width="58.5" height="15.75" align-x="186.75" align-y="2475.75" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:lessThan/> <ml:apply> <ml:lessThan/> <ml:real>0</ml:real> <ml:id xml:space="preserve">x</ml:id> </ml:apply> <ml:real>0.9</ml:real> </ml:apply> </math> <rendering item-idref="38"/> </region> <f family="Arial Cyr" charset="204" size="12">. Графическая иллюстрация полученного</f> <f family="Arial Cyr" charset="204" size="12">результата дана на рис.2.<sp count="2"/>Для<sp count="2"/>-1<x<0 вопрос оценки остатка ряда остается открытым!</f> </text> </region> <region region-id="194" left="42" top="2526" width="433.5" height="149.25" align-x="42" align-y="2526" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="39"/> <rendering item-idref="40"/> </region> <region region-id="138" left="42" top="2712.75" width="476.25" height="27" align-x="42" align-y="2724" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Полученное приближение намного точнее расчётного (тем более для </f> <f family="Arial Cyr" charset="204" size="12">мажоранты). Его </f> <f family="Arial Cyr" charset="204" size="12">следует показать отдельно на графике рис.3 и 4.</f> </text> </region> <region region-id="148" left="42" top="2766.75" width="123" height="15.75" align-x="87" align-y="2778" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve">E</ml:id> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">N</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:minus/> <ml:apply> <ml:id xml:space="preserve">f</ml:id> <ml:id xml:space="preserve">x</ml:id> </ml:apply> <ml:apply> <ml:id xml:space="preserve">S</ml:id> <ml:sequence> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">N</ml:id> </ml:sequence> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="41"/> </region> <region region-id="157" left="192" top="2766.75" width="112.5" height="15.75" align-x="261.75" align-y="2778" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:id xml:space="preserve">E</ml:id> <ml:sequence> <ml:real>-0.9</ml:real> <ml:real>154</ml:real> </ml:sequence> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>0.011106460996757051</ml:real> </result> </ml:eval> </math> <rendering item-idref="42"/> </region> <region region-id="196" left="324" top="2761.5" width="150" height="21" align-x="387" align-y="2778" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:id xml:space="preserve">E</ml:id> <ml:sequence> <ml:real>0.9</ml:real> <ml:real>154</ml:real> </ml:sequence> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>-0.00051783591292875464</ml:real> </result> </ml:eval> </math> <rendering item-idref="43"/> </region> <region region-id="195" left="36" top="2820" width="432.75" height="278.25" align-x="36" align-y="2820" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="44"/> <rendering item-idref="45"/> </region> <region region-id="256" left="48" top="3144" width="420.75" height="248.25" align-x="48" align-y="3144" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="46"/> <rendering item-idref="47"/> </region> <region region-id="199" left="66" top="3426.75" width="420" height="61.5" align-x="66" align-y="3438" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Графики E(x,N) показывают, что точность приближения сильно зависит от </f> <f family="Arial Cyr" charset="204" size="12">х. Но всегда можно выбрать такое значение </f> <region region-id="200" left="311.25" top="3441" width="27.75" height="16.5" align-x="320.25" align-y="3453" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">N</ml:id> <ml:id xml:space="preserve">ε</ml:id> </ml:apply> </math> <rendering item-idref="48"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">, что при n>N </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">будет выполняться </inlineAttr> </f> <region region-id="201" left="174.75" top="3458.25" width="101.25" height="16.5" align-x="260.25" align-y="3470.25" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:lessThan/> <ml:apply> <ml:absval/> <ml:apply> <ml:minus/> <ml:apply> <ml:id xml:space="preserve">f</ml:id> <ml:id xml:space="preserve">x</ml:id> </ml:apply> <ml:apply> <ml:id xml:space="preserve">S</ml:id> <ml:sequence> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">N</ml:id> </ml:sequence> </ml:apply> </ml:apply> </ml:apply> <ml:id xml:space="preserve">ε</ml:id> </ml:apply> </math> <rendering item-idref="49"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <sp/> </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> для всех </f> <region region-id="202" left="333" top="3459" width="38.25" height="15.75" align-x="356.25" align-y="3470.25" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:lessThan/> <ml:apply> <ml:absval/> <ml:id xml:space="preserve">x</ml:id> </ml:apply> <ml:real>1</ml:real> </ml:apply> </math> <rendering item-idref="50"/> </region> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">. Это свойство равномерной сходимости<sp count="2"/>также следует из графиков рис. 3 и 4.</inlineAttr> </f> </text> </region> <region region-id="177" left="78" top="3516.75" width="219" height="13.5" align-x="78" align-y="3528" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">Ниже приводятся данные для сранения:</f> </text> </region> <region region-id="176" left="78" top="3553.5" width="156" height="21" align-x="153.75" align-y="3570" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:id xml:space="preserve">E</ml:id> <ml:sequence> <ml:real>-0.85</ml:real> <ml:real>154</ml:real> </ml:sequence> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>1.0077544629893964E-06</ml:real> </result> </ml:eval> </math> <rendering item-idref="51"/> </region> <region region-id="173" left="288" top="3553.5" width="156" height="21" align-x="357" align-y="3570" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:id xml:space="preserve">E</ml:id> <ml:sequence> <ml:real>0.85</ml:real> <ml:real>154</ml:real> </ml:sequence> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>-7.5543619304507814E-08</ml:real> </result> </ml:eval> </math> <rendering item-idref="52"/> </region> <region region-id="175" left="78" top="3595.5" width="155.25" height="21" align-x="147.75" align-y="3612" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:id xml:space="preserve">E</ml:id> <ml:sequence> <ml:real>-0.8</ml:real> <ml:real>154</ml:real> </ml:sequence> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>6.127720553195104E-11</ml:real> </result> </ml:eval> </math> <rendering item-idref="53"/> </region> <region region-id="172" left="288" top="3595.5" width="155.25" height="21" align-x="351" align-y="3612" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:id xml:space="preserve">E</ml:id> <ml:sequence> <ml:real>0.8</ml:real> <ml:real>154</ml:real> </ml:sequence> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>-6.4458993698224276E-12</ml:real> </result> </ml:eval> </math> <rendering item-idref="54"/> </region> <region region-id="174" left="78" top="3631.5" width="155.25" height="21" align-x="147.75" align-y="3648" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:id xml:space="preserve">E</ml:id> <ml:sequence> <ml:real>-0.7</ml:real> <ml:real>154</ml:real> </ml:sequence> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>2.8421709430404007E-14</ml:real> </result> </ml:eval> </math> <rendering item-idref="55"/> </region> <region region-id="171" left="288" top="3636.75" width="84.75" height="15.75" align-x="351" align-y="3648" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:id xml:space="preserve">E</ml:id> <ml:sequence> <ml:real>0.7</ml:real> <ml:real>154</ml:real> </ml:sequence> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>-8.3266726846886741E-17</ml:real> </result> </ml:eval> </math> <rendering item-idref="56"/> </region> <region region-id="187" left="66" top="3672" width="221.25" height="16.5" align-x="66" align-y="3684" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Для значений </f> <region region-id="181" left="141.75" top="3672.75" width="69" height="15.75" align-x="195.75" align-y="3684" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:lessThan/> <ml:apply> <ml:lessThan/> <ml:real>0.9</ml:real> <ml:apply> 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top="3702.75" width="106.5" height="15.75" align-x="357" align-y="3714" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:id xml:space="preserve">E</ml:id> <ml:sequence> <ml:real>0.95</ml:real> <ml:real>154</ml:real> </ml:sequence> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>-2.1999148483361424</ml:real> </result> </ml:eval> </math> <rendering item-idref="59"/> </region> <region region-id="186" left="60" top="3738.75" width="440.25" height="27" align-x="60" align-y="3750" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Следовательно, в последнем случае количество членов ряда<sp count="2"/>S(x,N) надо брать 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