Question

Show that $\int_0^1\frac{\ln(1-x)}{1+x^2}\mathrm{d}x+\int_0^1\frac{\arctan x}{x(1+x)}\mathrm{d}x=0$

score 140 · Answer 1 · 2022-05-31 04:15:44

140

Views

Show that $\int_0^1\frac{\ln(1-x)}{1+x^2}\mathrm{d}x+\int_0^1\frac{\arctan x}{x(1+x)}\mathrm{d}x=0$

Published on 31 May 2022 - 4:15

score 112 · Answer 2 · 2022-06-01 02:01:29

112

Views

How to find $ \sum _{n=1}^{\infty} (-1)^n \frac{ \left(H_{n-1}-H_{\left\lfloor \frac{n-1}{2}\right\rfloor }\right)}{2n^3} $

Published on 01 Jun 2022 - 2:01

score 434 · Answer 3 · 2026-03-29 03:36:18

434

Views

Is the closed form of $\int_0^1\frac{\text{Li}_{2a+1}(x)}{1+x^2}dx$ known in the literature?

Published on 29 Mar 2026 - 3:36

score 195 · Answer 4 · 2022-06-15 16:00:56

195

Views

Is it possible to expand $\arctan x \ln(1-x^2)$ in series?

Published on 15 Jun 2022 - 16:00

score 313 · Answer 5 · 2022-06-23 08:18:56

313

Views

Closed form for $\sum_{k=1}^\infty\frac{H_k^{(m)}}{k^n}$

Published on 23 Jun 2022 - 8:18

score 115 · Answer 6 · 2026-02-23 12:08:06

115

Views

Closed form of $\sum _{k\ge 1} \frac{(-1)^{\binom{k}{p}}}{k}$, an alternating harmonic series with the signs determined by a binomial coeffcient

Published on 23 Feb 2026 - 12:08

score 116 · Answer 7 · 2022-07-10 12:28:59

116

Views

Finding a simpler form for a sum involving harmonic numbers and binomial coefficients

Published on 10 Jul 2022 - 12:28

score 350 · Answer 8 · 2022-07-14 19:22:13

350

Views

Integrate $\ln^3(\sin(x))$ using Fourier series

Published on 14 Jul 2022 - 19:22

score 78 · Answer 9 · 2022-07-17 21:50:54

78

Views

Sum of harmonic numbers inequality

Published on 17 Jul 2022 - 21:50

score 64 · Answer 10 · 2022-07-21 01:16:05

64

Views

A unspotted mistake involving simple Harmonic numbers

Published on 21 Jul 2022 - 1:16

score 46 · Answer 11 · 2022-08-13 17:59:36

46

Views

How to get a closed form for the sum $\sum_{n=1}^{\infty}{\frac{H_{2n}}{2n^2}}$?

Published on 13 Aug 2022 - 17:59

score 149 · Answer 12 · 2022-08-14 10:28:25

149

Views

Find a closed form for the sum with harmonic number $\sum_{n=0}^\infty (-1)^{n}\frac{H_{2n+1}^{2}}{(2n+1)^2}$

Published on 14 Aug 2022 - 10:28

score 96 · Answer 13 · 2022-09-08 22:20:52

96

Views

How do I use the following difference root (LDE) to extract a partial sum of the harmonic series below?

Published on 08 Sep 2022 - 22:20

score 139 · Answer 14 · 2026-03-27 05:41:58

139

Views

Deriving $\gamma \approx H(n)-\ln(n+1)+\frac{1}{2(n+1)}+\frac{1}{12(n+1)^2}$

Published on 27 Mar 2026 - 5:41

score 100 · Answer 15 · 2026-03-27 05:36:09

100

Views

How do I prove this Euler-Mascheroni limit?

Published on 27 Mar 2026 - 5:36

List Question

Show that $\int_0^1\frac{\ln(1-x)}{1+x^2}\mathrm{d}x+\int_0^1\frac{\arctan x}{x(1+x)}\mathrm{d}x=0$

How to find $ \sum _{n=1}^{\infty} (-1)^n \frac{ \left(H_{n-1}-H_{\left\lfloor \frac{n-1}{2}\right\rfloor }\right)}{2n^3} $

Is the closed form of $\int_0^1\frac{\text{Li}_{2a+1}(x)}{1+x^2}dx$ known in the literature?

Is it possible to expand $\arctan x \ln(1-x^2)$ in series?

Closed form for $\sum_{k=1}^\infty\frac{H_k^{(m)}}{k^n}$

Closed form of $\sum _{k\ge 1} \frac{(-1)^{\binom{k}{p}}}{k}$, an alternating harmonic series with the signs determined by a binomial coeffcient

Finding a simpler form for a sum involving harmonic numbers and binomial coefficients

Integrate $\ln^3(\sin(x))$ using Fourier series

Sum of harmonic numbers inequality

A unspotted mistake involving simple Harmonic numbers

How to get a closed form for the sum $\sum_{n=1}^{\infty}{\frac{H_{2n}}{2n^2}}$?

Find a closed form for the sum with harmonic number $\sum_{n=0}^\infty (-1)^{n}\frac{H_{2n+1}^{2}}{(2n+1)^2}$

How do I use the following difference root (LDE) to extract a partial sum of the harmonic series below?

Deriving $\gamma \approx H(n)-\ln(n+1)+\frac{1}{2(n+1)}+\frac{1}{12(n+1)^2}$

How do I prove this Euler-Mascheroni limit?

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