Problem 45831. Compute the Bernoulli numbers

Created by ChrisR

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<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>

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Last Solution submitted on May 31, 2022

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Problem 45831. Compute the Bernoulli numbers

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Problem 45831. Compute the Bernoulli numbers

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