Question

Approaching $\sum_{n=1}^\infty\frac{\overline{H}_n-H_{n/2}}{n^3}$ elegantly

score 204 · Answer 1 · 2026-03-26 13:02:02

204

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Approaching $\sum_{n=1}^\infty\frac{\overline{H}_n-H_{n/2}}{n^3}$ elegantly

Published on 26 Mar 2026 - 13:02

score 138 · Answer 2 · 2020-03-18 11:13:08

138

Views

Is there a way to express the inverse of harmonic numbers $1/H_k$ as a series

Published on 18 Mar 2020 - 11:13

score 263 · Answer 3 · 2020-03-18 21:35:57

263

Views

Compute $\int_0^1\frac{\ln(1-x)\ln(1+x)}{1+x}\ln\left(\frac{1+x}{2}\right)\ dx$

Published on 18 Mar 2020 - 21:35

score 280 · Answer 4 · 2020-03-19 18:49:30

280

Views

Nice pair of trilog integrals $\int_0^z \frac{\log ^2(x) \log (1\pm x)}{1\mp x} \, dx$

Published on 19 Mar 2020 - 18:49

score 1.1k · Answer 5 · 2020-03-20 14:44:17

1.1k

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Is this a valid proof that the harmonic series diverges?

Published on 20 Mar 2020 - 14:44

score 171 · Answer 6 · 2020-03-20 16:16:07

171

Views

Two unequal trilog integral twins $\int_0^1 \log^2(x)\frac{ \log(1\pm \sigma x)}{1\mp x}\,dx$

Published on 20 Mar 2020 - 16:16

score 163 · Answer 7 · 2020-03-23 11:02:20

163

Views

Limit of $\int_0^1 \frac{x^n-x^{2n}}{1-x}\text{d}x$

Published on 23 Mar 2020 - 11:02

score 65 · Answer 8 · 2020-03-26 11:56:52

65

Views

Prove that these two series are equal

Published on 26 Mar 2020 - 11:56

score 137 · Answer 9 · 2026-03-26 13:02:14

137

Views

Different approach to compute $\int_0^1\frac{\ln(x)}{1+x}\text{Li}_2\left(\frac{1+x}2\right)\ dx$

Published on 26 Mar 2026 - 13:02

score 186 · Answer 10 · 2020-03-27 16:09:12

186

Views

Find the limit of an infinite product related to the alternating harmonic convergence.

Published on 27 Mar 2020 - 16:09

score 131 · Answer 11 · 2026-03-25 09:29:03

131

Views

Prove $\lim_{x\to\infty}\sum_{n=1}^x x\log\left(1+\frac1{xn(an+1)}\right)= H_{\frac1a}$

Published on 25 Mar 2026 - 9:29

score 138 · Answer 12 · 2026-03-25 11:27:14

138

Views

Prove that the lower bound for the partial sums of the Möbius inverse of the Harmonic numbers minus $n$ is greater than $1$ minus a Harmonic number

Published on 25 Mar 2026 - 11:27

score 48 · Answer 13 · 2020-03-29 06:42:45

48

Views

Where does this asymptote for $H_n^{(k)}$ come from?

Published on 29 Mar 2020 - 6:42

score 685 · Answer 14 · 2020-03-29 12:43:35

685

Views

Integrating $\int_{0}^{1} \frac{\ln(1-x)}{x^2 + 1}$ via substitution and then Feynman's Technique

Published on 29 Mar 2020 - 12:43

score 709 · Answer 15 · 2020-03-31 08:09:13

709

Views

A suprising conjectural closed-form of $\sum _{n=1}^{\infty } \frac{1}{n^4 2^n \binom{3 n}{n}}$ and integral variations

Published on 31 Mar 2020 - 8:09

List Question

Approaching $\sum_{n=1}^\infty\frac{\overline{H}_n-H_{n/2}}{n^3}$ elegantly

Is there a way to express the inverse of harmonic numbers $1/H_k$ as a series

Compute $\int_0^1\frac{\ln(1-x)\ln(1+x)}{1+x}\ln\left(\frac{1+x}{2}\right)\ dx$

Nice pair of trilog integrals $\int_0^z \frac{\log ^2(x) \log (1\pm x)}{1\mp x} \, dx$

Is this a valid proof that the harmonic series diverges?

Two unequal trilog integral twins $\int_0^1 \log^2(x)\frac{ \log(1\pm \sigma x)}{1\mp x}\,dx$

Limit of $\int_0^1 \frac{x^n-x^{2n}}{1-x}\text{d}x$

Prove that these two series are equal

Different approach to compute $\int_0^1\frac{\ln(x)}{1+x}\text{Li}_2\left(\frac{1+x}2\right)\ dx$

Find the limit of an infinite product related to the alternating harmonic convergence.

Prove $\lim_{x\to\infty}\sum_{n=1}^x x\log\left(1+\frac1{xn(an+1)}\right)= H_{\frac1a}$

Prove that the lower bound for the partial sums of the Möbius inverse of the Harmonic numbers minus $n$ is greater than $1$ minus a Harmonic number

Where does this asymptote for $H_n^{(k)}$ come from?

Integrating $\int_{0}^{1} \frac{\ln(1-x)}{x^2 + 1}$ via substitution and then Feynman's Technique

A suprising conjectural closed-form of $\sum _{n=1}^{\infty } \frac{1}{n^4 2^n \binom{3 n}{n}}$ and integral variations

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