Problem 45831. Compute the Bernoulli numbers

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>The</w:t></w:r><w:r><w:t> </w:t></w:r><w:hyperlink w:docLocation=\"https://en.wikipedia.org/wiki/Bernoulli_number\"><w:r><w:t>Bernoulli numbers</w:t></w:r></w:hyperlink><w:r><w:t> appear in several applications in number theory, including expressions for sums of powers and certain values of the Riemann zeta function. For each index </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=\"m\"/></w:customXmlPr><w:r><w:t>m</w:t></w:r></w:customXml><w:r><w:t>, the Bernoulli numbers have two values, one denoted with a plus superscript and one denoted with a minus superscript.</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 </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=\"m\"/></w:customXmlPr><w:r><w:t>m</w:t></w:r></w:customXml><w:r><w:t> = 0, 1, 2, 3, 4, the \"plus\" values are </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=\"B_m^+\"/></w:customXmlPr><w:r><w:t>B_m^+</w:t></w:r></w:customXml><w:r><w:t> = 1, 1/2, 1/6, 0, and -1/30, respectively. The \"minus\" values are the same, except for </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=\"m\"/></w:customXmlPr><w:r><w:t>m</w:t></w:r></w:customXml><w:r><w:t> = 1, for which the value is -1/2.</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>Compute the two values of the Bernoulli numbers for the given index. Round to seven places after the decimal.</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: 146.817px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 73.4167px; transform-origin: 407px 73.4167px; vertical-align: baseline; "><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: 12.0667px 7.8px; transform-origin: 12.0667px 7.8px; unicode-bidi: normal; "><span style="">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/Bernoulli_number"><span style=""><span style="">Bernoulli numbers</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: 295.917px 7.8px; transform-origin: 295.917px 7.8px; unicode-bidi: normal; "><span style=""> appear in several applications in number theory, including expressions for sums of powers and certain values of the Riemann zeta function. For each index </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);">m</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: 189.05px 7.8px; transform-origin: 189.05px 7.8px; unicode-bidi: normal; "><span style="">, the Bernoulli numbers have two values, one denoted with a plus superscript and one denoted with a minus superscript.</span></span></div><div style="block-size: 44.8167px; 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 22.4167px; text-align: left; transform-origin: 384px 22.4167px; 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: 12.45px 7.8px; transform-origin: 12.45px 7.8px; unicode-bidi: normal; "><span style="">For </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);">m</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: 112.5px 7.8px; transform-origin: 112.5px 7.8px; unicode-bidi: normal; "><span style=""> = 0, 1, 2, 3, 4, the "plus" values are </span></span><span style="vertical-align:-8px"><img src="data:image/png;base64,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" alt="B_m^+" style="width: 20px; height: 22px;" width="20" height="22"></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: 231px 7.8px; transform-origin: 231px 7.8px; unicode-bidi: normal; "><span style=""> = 1, 1/2, 1/6, 0, and -1/30, respectively. The "minus" values are the same, except for </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);">m</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: 97.05px 7.8px; transform-origin: 97.05px 7.8px; unicode-bidi: normal; "><span style=""> = 1, for which the value is -1/2.</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: 342.7px 7.8px; transform-origin: 342.7px 7.8px; unicode-bidi: normal; "><span style="">Compute the two values of the Bernoulli numbers for the given index. Round to seven places after the decimal.</span></span></div></div></div>

Solve

Solution Stats

31 Solutions
8 Solvers

Last Solution submitted on Feb 02, 2022

Last 200 Solutions

Problem Recent Solvers8

Problem Tags

number theory

Community Treasure Hunt

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

Start Hunting!

Problem 45831. Compute the Bernoulli numbers

Solution Stats

Last 200 Solutions

Problem Recent Solvers8

Suggested Problems

More from this Author147

Problem Tags

Community Treasure Hunt

Problem 45831. Compute the Bernoulli numbers

Solution Stats

Last 200 Solutions

Problem Recent Solvers8

Suggested Problems

More from this Author147

Problem Tags

Community Treasure Hunt

Players