Problem 45939. Estimate π from certain values of the zeta function

Created by ChrisR

Solve Later

{"parts":[{"partUri":"/matlab/document.xml","contentType":"application/vnd.mathworks.matlab.code.document+xml","content":"<?xml version=\"1.0\" encoding=\"UTF-8\"?><w:document xmlns:w=\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\"><w:body><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>Cody Problems</w:t></w:r><w:r><w:t> </w:t></w:r><w:hyperlink w:docLocation=\"https://www.mathworks.com/matlabcentral/cody/problems/2908-approximation-of-pi\"><w:r><w:t>2908</w:t></w:r></w:hyperlink><w:r><w:t> and</w:t></w:r><w:r><w:t> </w:t></w:r><w:hyperlink w:docLocation=\"https://www.mathworks.com/matlabcentral/cody/problems/2909-approximation-of-pi-vector-inputs\"><w:r><w:t>2909</w:t></w:r></w:hyperlink><w:r><w:t> ask us to estimate </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"pi\"/></w:customXmlPr><w:r><w:t>\\pi</w:t></w:r></w:customXml><w:r><w:t> by summing a given number of terms in an infinite sum, the Leibniz formula.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>Here you are asked to estimate </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"pi\"/></w:customXmlPr><w:r><w:t>\\pi</w:t></w:r></w:customXml><w:r><w:t> by summing a given number of terms in other infinite sums corresponding to the</w:t></w:r><w:r><w:t> </w:t></w:r><w:hyperlink w:docLocation=\"https://en.wikipedia.org/wiki/Riemann_zeta_function\"><w:r><w:t>Riemann zeta function</w:t></w:r></w:hyperlink><w:r><w:t> evaluated at positive even integers. For example, in solving the</w:t></w:r><w:r><w:t> </w:t></w:r><w:hyperlink w:docLocation=\"https://en.wikipedia.org/wiki/Basel_problem\"><w:r><w:t>Basel problem</w:t></w:r></w:hyperlink><w:r><w:t>, Euler summed the reciprocals of the squares of positive integers and showed that </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"zeta(2) = pi^2/6\"/></w:customXmlPr><w:r><w:t>\\zeta(2) = \\pi^2/6</w:t></w:r></w:customXml><w:r><w:t>.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>Write a function that takes a vector </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>n</w:t></w:r><w:r><w:t> with the numbers of terms to sum and a vector </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>m</w:t></w:r><w:r><w:t> with the values at which to estimate the zeta function. The function should return an array of character strings with the absolute value of the relative error of the estimate E--that is,</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>abs(E-pi)/pi</w:t></w:r><w:r><w:t>. Report the relative error in scientific notation using num2str(...,'%10.2e').</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>For example, PiByZeta(30,2) should return</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"code\"/></w:pPr><w:r><w:t><![CDATA[ '1.00e-02']]></w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>while PiByZeta([25 30],[2 8]) should return</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"code\"/></w:pPr><w:r><w:t><![CDATA[ '1.20e-02  1.00e-02'\n '2.53e-12  7.22e-13']]></w:t></w:r></w:p></w:body></w:document>","relationship":null}],"relationships":[{"relationshipType":"http://schemas.mathworks.com/matlab/code/2013/relationships/document","target":"/matlab/document.xml","relationshipId":"rId1"}]}

<div style = "text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; "><div style="block-size: 329.2px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 164.6px; transform-origin: 407px 164.6px; vertical-align: baseline; "><div style="block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 47.85px 7.8px; transform-origin: 47.85px 7.8px; unicode-bidi: normal; "><span style="">Cody Problems</span></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; "><span style=""> </span></span><a target='_blank' href = "https://www.mathworks.com/matlabcentral/cody/problems/2908-approximation-of-pi"><span style=""><span style="">2908</span></span></a><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 13.6167px 7.8px; transform-origin: 13.6167px 7.8px; unicode-bidi: normal; "><span style=""> and</span></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; "><span style=""> </span></span><a target='_blank' href = "https://www.mathworks.com/matlabcentral/cody/problems/2909-approximation-of-pi-vector-inputs"><span style=""><span style="">2909</span></span></a><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 60.2833px 7.8px; transform-origin: 60.2833px 7.8px; unicode-bidi: normal; "><span style=""> ask us to estimate </span></span><span style="font-family: &quot;STIXGeneral&quot;, &quot;STIXGeneral-webfont&quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);">π</span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 211.217px 7.8px; transform-origin: 211.217px 7.8px; unicode-bidi: normal; "><span style=""> by summing a given number of terms in an infinite sum, the Leibniz formula.</span></span></div><div style="block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 99.1833px 7.8px; transform-origin: 99.1833px 7.8px; unicode-bidi: normal; "><span style="">Here you are asked to estimate </span></span><span style="font-family: &quot;STIXGeneral&quot;, &quot;STIXGeneral-webfont&quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);">π</span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 248.95px 7.8px; transform-origin: 248.95px 7.8px; unicode-bidi: normal; "><span style=""> by summing a given number of terms in other infinite sums corresponding to the</span></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; "><span style=""> </span></span><a target='_blank' href = "https://en.wikipedia.org/wiki/Riemann_zeta_function"><span style=""><span style="perspective-origin: 69.6333px 7.8px; transform-origin: 69.6333px 7.8px; ">Riemann zeta function</span></span></a><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 197.217px 7.8px; transform-origin: 197.217px 7.8px; unicode-bidi: normal; "><span style=""> evaluated at positive even integers. For example, in solving the</span></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; "><span style=""> </span></span><a target='_blank' href = "https://en.wikipedia.org/wiki/Basel_problem"><span style=""><span style="">Basel problem</span></span></a><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 62.6167px 7.8px; transform-origin: 62.6167px 7.8px; unicode-bidi: normal; "><span style="">, Euler summed the reciprocals of the squares of positive integers and showed that </span></span><span style="vertical-align:-5px"><img src="data:image/png;base64,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" alt="zeta(2) = pi^2/6" style="width: 78px; height: 19.5px;" width="78" height="19.5"></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; "><span style="">.</span></span></div><div style="block-size: 63.9px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.95px; text-align: left; transform-origin: 384px 31.95px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 109.55px 7.8px; transform-origin: 109.55px 7.8px; unicode-bidi: normal; "><span style="">Write a function that takes a vector </span></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.85px 7.8px; transform-origin: 3.85px 7.8px; unicode-bidi: normal; "><span style="font-family: Menlo, Monaco, Consolas, &quot;Courier New&quot;, monospace; perspective-origin: 3.85px 8.25px; transform-origin: 3.85px 8.25px; ">n</span></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 147.783px 7.8px; transform-origin: 147.783px 7.8px; unicode-bidi: normal; "><span style=""> with the numbers of terms to sum and a vector </span></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.85px 7.8px; transform-origin: 3.85px 7.8px; unicode-bidi: normal; "><span style="font-family: Menlo, Monaco, Consolas, &quot;Courier New&quot;, monospace; perspective-origin: 3.85px 8.25px; transform-origin: 3.85px 8.25px; ">m</span></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 113.967px 7.8px; transform-origin: 113.967px 7.8px; unicode-bidi: normal; "><span style=""> with the values at which to estimate the zeta function. The function should return an array of character strings with the absolute value of the relative error of the estimate E--that is,</span></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; "><span style=""> </span></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 46.2px 7.8px; transform-origin: 46.2px 7.8px; unicode-bidi: normal; "><span style="font-family: Menlo, Monaco, Consolas, &quot;Courier New&quot;, monospace; perspective-origin: 46.2px 8.25px; transform-origin: 46.2px 8.25px; ">abs(E-pi)/pi</span></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 227.483px 7.8px; transform-origin: 227.483px 7.8px; unicode-bidi: normal; "><span style="">. Report the relative error in scientific notation using num2str(...,'%10.2e').</span></span></div><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 132.25px 7.8px; transform-origin: 132.25px 7.8px; unicode-bidi: normal; "><span style="">For example, PiByZeta(30,2) should return</span></span></div><div style="background-color: rgb(247, 247, 247); block-size: 20.4333px; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-end-end-radius: 4px; border-end-start-radius: 4px; border-start-end-radius: 4px; border-start-start-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; margin-block-end: 10px; margin-block-start: 10px; margin-bottom: 10px; margin-inline-end: 3px; margin-inline-start: 3px; margin-left: 3px; margin-right: 3px; margin-top: 10px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; margin-left: 3px; margin-top: 10px; margin-bottom: 10px; margin-right: 3px; "><div style="background-color: rgba(0, 0, 0, 0); border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.916667px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.916667px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.916667px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.916667px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; white-space: nowrap; "><span style="block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 42.35px 8.25px; transform-origin: 42.35px 8.25px; unicode-bidi: normal; white-space: pre; margin-right: 45px; "><span style="margin-inline-end: 0px; margin-right: 0px; perspective-origin: 3.85px 8.25px; transform-origin: 3.85px 8.25px; "> </span><span style="border-block-end-color: rgb(170, 4, 249); border-block-start-color: rgb(170, 4, 249); border-bottom-color: rgb(170, 4, 249); border-inline-end-color: rgb(170, 4, 249); border-inline-start-color: rgb(170, 4, 249); border-left-color: rgb(170, 4, 249); border-right-color: rgb(170, 4, 249); border-top-color: rgb(170, 4, 249); caret-color: rgb(170, 4, 249); color: rgb(170, 4, 249); column-rule-color: rgb(170, 4, 249); margin-inline-end: 0px; margin-right: 0px; outline-color: rgb(170, 4, 249); perspective-origin: 38.5px 8.25px; text-decoration: none; text-decoration-color: rgb(170, 4, 249); text-emphasis-color: rgb(170, 4, 249); transform-origin: 38.5px 8.25px; ">'1.00e-02'</span></span></div></div><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 10px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 10px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 10px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 130.7px 7.8px; transform-origin: 130.7px 7.8px; unicode-bidi: normal; "><span style="">while PiByZeta([25 30],[2 8]) should return</span></span></div><div style="background-color: rgb(247, 247, 247); block-size: 40.8667px; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-end-end-radius: 4px; border-end-start-radius: 4px; border-start-end-radius: 4px; border-start-start-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; margin-block-end: 10px; margin-block-start: 10px; margin-bottom: 10px; margin-inline-end: 3px; margin-inline-start: 3px; margin-left: 3px; margin-right: 3px; margin-top: 10px; perspective-origin: 404px 20.4333px; transform-origin: 404px 20.4333px; margin-left: 3px; margin-top: 10px; margin-bottom: 10px; margin-right: 3px; "><div style="background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.916667px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.916667px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.916667px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.916667px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; "><span style="block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 80.85px 8.25px; transform-origin: 80.85px 8.25px; unicode-bidi: normal; white-space: pre; margin-right: 45px; "><span style="margin-inline-end: 0px; margin-right: 0px; perspective-origin: 3.85px 8.25px; transform-origin: 3.85px 8.25px; "> </span><span style="border-block-end-color: rgb(170, 4, 249); border-block-start-color: rgb(170, 4, 249); border-bottom-color: rgb(170, 4, 249); border-inline-end-color: rgb(170, 4, 249); border-inline-start-color: rgb(170, 4, 249); border-left-color: rgb(170, 4, 249); border-right-color: rgb(170, 4, 249); border-top-color: rgb(170, 4, 249); caret-color: rgb(170, 4, 249); color: rgb(170, 4, 249); column-rule-color: rgb(170, 4, 249); margin-inline-end: 0px; margin-right: 0px; outline-color: rgb(170, 4, 249); perspective-origin: 77px 8.25px; text-decoration: none; text-decoration-color: rgb(170, 4, 249); text-emphasis-color: rgb(170, 4, 249); transform-origin: 77px 8.25px; ">'1.20e-02  1.00e-02'</span></span></div><div style="background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.916667px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.916667px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.916667px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.916667px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; "><span style="block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 80.85px 8.25px; transform-origin: 80.85px 8.25px; unicode-bidi: normal; white-space: pre; margin-right: 45px; "><span style="margin-inline-end: 0px; margin-right: 0px; perspective-origin: 3.85px 8.25px; transform-origin: 3.85px 8.25px; "> </span><span style="border-block-end-color: rgb(170, 4, 249); border-block-start-color: rgb(170, 4, 249); border-bottom-color: rgb(170, 4, 249); border-inline-end-color: rgb(170, 4, 249); border-inline-start-color: rgb(170, 4, 249); border-left-color: rgb(170, 4, 249); border-right-color: rgb(170, 4, 249); border-top-color: rgb(170, 4, 249); caret-color: rgb(170, 4, 249); color: rgb(170, 4, 249); column-rule-color: rgb(170, 4, 249); margin-inline-end: 0px; margin-right: 0px; outline-color: rgb(170, 4, 249); perspective-origin: 77px 8.25px; text-decoration: none; text-decoration-color: rgb(170, 4, 249); text-emphasis-color: rgb(170, 4, 249); transform-origin: 77px 8.25px; ">'2.53e-12  7.22e-13'</span></span></div></div></div></div>

Cody Problems <https://www.mathworks.com/matlabcentral/cody/problems/2908-approximation-of-pi 2908> and <https://www.mathworks.com/matlabcentral/cody/problems/2909-approximation-of-pi-vector-inputs 2909> ask us to estimate π by summing a given number of terms in an infinite sum, the Leibniz formula.

Here you are asked to estimate π by summing a given number of terms in other infinite sums corresponding to the <https://en.wikipedia.org/wiki/Riemann_zeta_function Riemann zeta function> evaluated at positive even integers. For example, in solving the <https://en.wikipedia.org/wiki/Basel_problem Basel problem>, Euler summed the reciprocals of the squares of positive integers and showed that ζ(2) = π^2/6.

Write a function that takes a vector n with the numbers of terms to sum and a vector m with the values at which to estimate the zeta function. The function should return an array of character strings with the absolute value of the relative error of the estimate E--that is, |abs(E-pi)/pi|. Report the relative error in scientific notation using num2str(...,'%10.2e').

For example, PiByZeta(30,2) should return

'1.00e-02'
 
while PiByZeta([25 30],[2 8]) should return

'1.20e-02  1.00e-02'
 '2.53e-12  7.22e-13'

Solve

Solution Stats

13 Solutions
6 Solvers

Last Solution submitted on Jun 19, 2022

Last 200 Solutions

Problem Recent Solvers6

Problem Tags

number theory pi zeta function

Community Treasure Hunt

Find the treasures in MATLAB Central and discover how the community can help you!

Start Hunting!

Problem 45939. Estimate π from certain values of the zeta function

Solution Stats

Last 200 Solutions

Problem Recent Solvers6

Suggested Problems

More from this Author156

Problem Tags

Community Treasure Hunt

Problem 45939. Estimate π from certain values of the zeta function

Solution Stats

Last 200 Solutions

Problem Recent Solvers6

Suggested Problems

More from this Author156

Problem Tags

Community Treasure Hunt

Players