Problem 46000. Compute the harmonic 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: 210px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 105px; transform-origin: 407px 105px; vertical-align: baseline; "><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: 14px 7.8px; transform-origin: 14px 7.8px; unicode-bidi: normal; "><span style="">The </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);">n</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: 5.83333px 7.8px; transform-origin: 5.83333px 7.8px; unicode-bidi: normal; "><span style="">th</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://mathworld.wolfram.com/HarmonicNumber.html"><span style=""><span style="">harmonic number</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: 201.083px 7.8px; transform-origin: 201.083px 7.8px; unicode-bidi: normal; "><span style=""> is defined as the sum of the reciprocals of the integers from 1 to </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);">n</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: 45px; 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 22.5px; text-align: left; transform-origin: 384px 22.5px; 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: 13.6px 7.8px; transform-origin: 13.6px 7.8px; unicode-bidi: normal; "><span style="">       </span></span><span style="vertical-align:-17px"><img src="data:image/png;base64,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" alt="Definition of the harmonic numbers: sum(1/k) from k = 1 to k = n" style="width: 69px; height: 45px;" width="69" height="45"></span></div><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: 381.85px 7.8px; transform-origin: 381.85px 7.8px; unicode-bidi: normal; "><span style="">The first four harmonic numbers are 1, 3/2, 11/6, and 25/12. The harmonic numbers appear in several interesting identities, including this one for</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/Ap%C3%A9ry's_constant"><span style=""><span style="">Apery's constant</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: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; "><span style="">:</span></span></div><div style="block-size: 45px; 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 22.5px; text-align: left; transform-origin: 384px 22.5px; 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: 13.6px 7.8px; transform-origin: 13.6px 7.8px; unicode-bidi: normal; "><span style="">       </span></span><span style="vertical-align:-17px"><img src="data:image/png;base64,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" alt="Relation between zeta(3) and the harmonic numbers" style="width: 98.5px; height: 45px;" width="98.5" height="45"></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: 42.0167px 7.8px; transform-origin: 42.0167px 7.8px; unicode-bidi: normal; "><span style="">Compute the </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);">n</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: 63.8px 7.8px; transform-origin: 63.8px 7.8px; unicode-bidi: normal; "><span style="">th harmonic number.</span></span></div></div></div>

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Last Solution submitted on Jun 27, 2022

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1 Comment

Binbin Qi on 18 Nov 2020

There is something wrong. The solution I submit is not the solution I edited instead of original file.

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Solution 2631952

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ChrisR on 1 Jul 2020

I was intrigued by this identity, but the straightforward approach scores better in Cody.

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Problem 46000. Compute the harmonic numbers

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Problem 46000. Compute the harmonic numbers

Solution Stats

Last 200 Solutions

Problem Comments

Solution Comments

Problem Recent Solvers44

Suggested Problems

More from this Author157

Problem Tags

Community Treasure Hunt

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