List questions from harmonic-numbers

Published on 17 Apr 2026 - 13:24

323

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Closed form for $1+\frac{1}{2} + \frac{1}{3} + \dots + \frac{1}{n}$?

Published on 15 Apr 2026 - 13:38

47

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This semi-harmonic-series converges

Published on 13 Apr 2026 - 22:18

130

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Evaluating the continued fraction

Published on 10 May 2026 - 22:08

509

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Evaluating $1-\frac12-\frac13+\frac14+\frac15+\frac16-\cdots$

Published on 14 Apr 2026 - 17:52

3.6k

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Proving that $\int_0^1 \frac{\arctan x}{x}\ln\left(\frac{1+x^2}{(1-x)^2}\right)dx=\frac{\pi^3}{16}$

Published on 17 Apr 2026 - 5:29

62

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Isolating $k$ in $H_k=\frac{c}{k+1}$

Published on 09 Apr 2026 - 21:13

222

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Showing $\sum_{k=1}^{nm} \frac{1}{k} \approx \sum_{k=1}^{n} \frac{1}{k} + \sum_{k=1}^{m} \frac{1}{k}$

Published on 15 Apr 2026 - 18:39

2k

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Analytic continuation of harmonic series

Published on 10 May 2026 - 19:10

150

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Closed expression for alternating harmonically wrapped harmonic series $\sum_{n=1}^\infty (-1)^{n+1} H_{\frac{1}{n}}$

Published on 13 Apr 2026 - 3:26

263

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Generalizing $\sum_{n=1}^{\infty}\frac{H_n{2n \choose n}}{2^{2n}(2n-1)}=2$

Published on 15 Apr 2026 - 18:14

76

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Is there a way to sum (part of) the harmonic series to a given total?

Published on 11 Apr 2026 - 23:19

257

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Logarithmic integral $ \int_0^1 \frac{x}{x^2+1} \, \log(x)\log(x+1) \, {\rm d}x $

Published on 10 May 2026 - 15:46

278

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How to show that $\sum_{n=1}^{\infty}\frac{H_n}{n^2+n}=\frac{\pi^2}{6}$

Published on 13 Apr 2026 - 2:05

95

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Proof of harmonic series divergence

Published on 11 Apr 2026 - 12:34

List Question

Is $1 + \frac{1}{2} + \frac{1}{3} + \dots + \frac{1}{x} < \log_2 x$

Closed form for $1+\frac{1}{2} + \frac{1}{3} + \dots + \frac{1}{n}$?

This semi-harmonic-series converges

Evaluating the continued fraction

Evaluating $1-\frac12-\frac13+\frac14+\frac15+\frac16-\cdots$

Proving that $\int_0^1 \frac{\arctan x}{x}\ln\left(\frac{1+x^2}{(1-x)^2}\right)dx=\frac{\pi^3}{16}$

Isolating $k$ in $H_k=\frac{c}{k+1}$

Showing $\sum_{k=1}^{nm} \frac{1}{k} \approx \sum_{k=1}^{n} \frac{1}{k} + \sum_{k=1}^{m} \frac{1}{k}$

Analytic continuation of harmonic series

Closed expression for alternating harmonically wrapped harmonic series $\sum_{n=1}^\infty (-1)^{n+1} H_{\frac{1}{n}}$

Generalizing $\sum_{n=1}^{\infty}\frac{H_n{2n \choose n}}{2^{2n}(2n-1)}=2$

Is there a way to sum (part of) the harmonic series to a given total?

Logarithmic integral $ \int_0^1 \frac{x}{x^2+1} \, \log(x)\log(x+1) \, {\rm d}x $

How to show that $\sum_{n=1}^{\infty}\frac{H_n}{n^2+n}=\frac{\pi^2}{6}$

Proof of harmonic series divergence

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