Question

Prove $\displaystyle\sum_{-\infty}^0\text H(x)=\frac12\sum_0^\infty\frac{\text H_\frac x2-\text H_\frac{x-1}2}{x!}=\frac{\text{Ei}(2)-\text{Ei}(1)}e$

score 345 · Answer 1 · 2021-09-30 14:05:33

345

Views

Prove $\displaystyle\sum_{-\infty}^0\text H(x)=\frac12\sum_0^\infty\frac{\text H_\frac x2-\text H_\frac{x-1}2}{x!}=\frac{\text{Ei}(2)-\text{Ei}(1)}e$

Published on 30 Sep 2021 - 14:05

score 207 · Answer 2 · 2026-03-31 13:36:40

207

Views

Integer Values of $\sum_{k=1}^n k^r . \sum_{q=1}^n \frac{1}{q^r}$

Published on 31 Mar 2026 - 13:36

score 38 · Answer 3 · 2021-10-20 19:03:02

38

Views

Asymptotic behaviour of the $\sum_{k=3}^n \frac{n!}{(n-k)! n^k k}$

Published on 20 Oct 2021 - 19:03

score 101 · Answer 4 · 2021-10-31 01:07:48

101

Views

Hint for proving $H_n=\sum_{i=1}^n\binom{n}{i}\frac{(-1)^{i+1}}i$, where $H_i=\sum_{k=1}^i\frac1k$

Published on 31 Oct 2021 - 1:07

score 341 · Answer 5 · 2026-03-29 03:35:56

341

Views

Evaluating $\int_0^1\frac{\operatorname{Li}_2(x)\ln(1+x)}x\,dx$

Published on 29 Mar 2026 - 3:35

score 157 · Answer 6 · 2021-11-09 10:25:30

157

Views

nth term test for the series $\sum_{n=1}^{\infty }{1/n} $

Published on 09 Nov 2021 - 10:25

score 148 · Answer 7 · 2021-11-14 10:30:01

148

Views

The harmonic Series sequence.

Published on 14 Nov 2021 - 10:30

score 313 · Answer 8 · 2026-03-26 17:53:26

313

Views

Trying to prove $\left(1-\frac13+\frac15-\frac17+\cdots\right)^2=\frac38\left(\frac1{1^2}+\frac1{2^2}+\frac1{3^2}+\frac1{4^2}+\cdots\right)$

Published on 26 Mar 2026 - 17:53

score 162 · Answer 9 · 2021-11-24 04:35:30

162

Views

Is there a closed a form for $\sum_{n=1}^\infty\frac{(-1)^{n-1}H_{an}}{n}\,?$

Published on 24 Nov 2021 - 4:35

score 537 · Answer 10 · 2021-11-25 20:15:53

537

Views

Coefficient extraction,show: $[z^n]\frac{1}{(1-z)^{\alpha + 1}} \log \frac{1}{1-z} = \binom{n + \alpha }{n} (H_{n+\alpha} - H_{\alpha})$

Published on 25 Nov 2021 - 20:15

score 61 · Answer 11 · 2021-12-01 23:59:32

61

Views

Prove that this subserie of the harmonic series is convergent.

Published on 01 Dec 2021 - 23:59

score 216 · Answer 12 · 2026-03-25 07:15:32

216

Views

How do I evaluate $\sum_{n=1}^{\infty}\frac{(-1)^{n+1}H_n}{2n+1}?$

Published on 25 Mar 2026 - 7:15

score 330 · Answer 13 · 2026-03-29 03:35:40

330

Views

Different ways to evaluate $\sum_{n=1}^\infty\frac{H_nH_n^{(2)}}{(n+1)(n+2)(n+3)}$

Published on 29 Mar 2026 - 3:35

score 32 · Answer 14 · 2021-12-16 07:33:31

32

Views

An identity satisfied by 'harmonic numbers'

Published on 16 Dec 2021 - 7:33

score 546 · Answer 15 · 2026-02-23 15:04:01

546

Views

Find three integers in Harmonic Progression

Published on 23 Feb 2026 - 15:04

List Question

Prove $\displaystyle\sum_{-\infty}^0\text H(x)=\frac12\sum_0^\infty\frac{\text H_\frac x2-\text H_\frac{x-1}2}{x!}=\frac{\text{Ei}(2)-\text{Ei}(1)}e$

Integer Values of $\sum_{k=1}^n k^r . \sum_{q=1}^n \frac{1}{q^r}$

Asymptotic behaviour of the $\sum_{k=3}^n \frac{n!}{(n-k)! n^k k}$

Hint for proving $H_n=\sum_{i=1}^n\binom{n}{i}\frac{(-1)^{i+1}}i$, where $H_i=\sum_{k=1}^i\frac1k$

Evaluating $\int_0^1\frac{\operatorname{Li}_2(x)\ln(1+x)}x\,dx$

nth term test for the series $\sum_{n=1}^{\infty }{1/n} $

The harmonic Series sequence.

Trying to prove $\left(1-\frac13+\frac15-\frac17+\cdots\right)^2=\frac38\left(\frac1{1^2}+\frac1{2^2}+\frac1{3^2}+\frac1{4^2}+\cdots\right)$

Is there a closed a form for $\sum_{n=1}^\infty\frac{(-1)^{n-1}H_{an}}{n}\,?$

Coefficient extraction,show: $[z^n]\frac{1}{(1-z)^{\alpha + 1}} \log \frac{1}{1-z} = \binom{n + \alpha }{n} (H_{n+\alpha} - H_{\alpha})$

Prove that this subserie of the harmonic series is convergent.

How do I evaluate $\sum_{n=1}^{\infty}\frac{(-1)^{n+1}H_n}{2n+1}?$

Different ways to evaluate $\sum_{n=1}^\infty\frac{H_nH_n^{(2)}}{(n+1)(n+2)(n+3)}$

An identity satisfied by 'harmonic numbers'

Find three integers in Harmonic Progression

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