Question

Calculating logarithmic integrals without using the derivatives of Beta function.

score 383 · Answer 1 · 2019-10-21 00:23:07

383

Views

Calculating logarithmic integrals without using the derivatives of Beta function.

Published on 21 Oct 2019 - 0:23

score 142 · Answer 2 · 2019-10-23 00:33:26

142

Views

Calulating $\int_0^1\frac{5\ln^4x\ln^4(1-x)-8\ln^5x\ln^3(1-x)+4\ln^2(1-x)\ln^6x}{x(1-x)}\ dx$

Published on 23 Oct 2019 - 0:33

score 170 · Answer 3 · 2019-10-24 06:11:07

170

Views

A bound of the harmonic series of squares.

Published on 24 Oct 2019 - 6:11

score 355 · Answer 4 · 2019-10-24 06:27:05

355

Views

Advanced : Compute $\sum_{n=1}^\infty\frac{H_n^4-6H_n^2H_n^{(2)}+8H_nH_n^{(3)}+3\left(H_n^{(2)}\right)^2-6H_n^{(4)}}{n^5}$

Published on 24 Oct 2019 - 6:27

score 101 · Answer 5 · 2019-10-24 13:59:46

101

Views

A group of advanced harmonic series with weight of 9

Published on 24 Oct 2019 - 13:59

score 394 · Answer 6 · 2026-03-26 04:36:24

394

Views

Is there a closed form for $\int_0^1\frac{\ln^4(1-x)\operatorname{Li}_4(x)}{x}dx\ ?$

Published on 26 Mar 2026 - 4:36

score 394 · Answer 7 · 2026-03-22 15:51:23

394

Views

Positivity of a certain sum of Stirling numbers

Published on 22 Mar 2026 - 15:51

score 175 · Answer 8 · 2019-11-01 01:18:25

175

Views

Closed form for $\int_{0}^1 \frac{\ln(x)\ln(1-x)^b}{x^2}dx$

Published on 01 Nov 2019 - 1:18

score 268 · Answer 9 · 2019-11-03 02:12:52

268

Views

Generalization of Harmonic series and Euler-Mascheroni Constant

Published on 03 Nov 2019 - 2:12

score 120 · Answer 10 · 2019-11-03 22:28:13

120

Views

Closed Form for $\int_{0}^1\frac{\ln(x)\ln^a(1-x)}{x^3}dx$

Published on 03 Nov 2019 - 22:28

score 373 · Answer 11 · 2019-11-04 22:20:42

373

Views

Finding the sum of $\sum^\infty_{k=1}{\frac{1}{k(k+2)}}$ with partial fraction decomposition

Published on 04 Nov 2019 - 22:20

score 805 · Answer 12 · 2019-11-07 02:00:02

805

Views

Prove $\zeta(3)=2\sum_{n=1}^\infty\frac{H_n}{n}\left[\frac1{4^n}{2n\choose n}\left(H_{2n}-H_n-\frac1{2n}-\ln2\right)+\frac1{2n}\right]$

Published on 07 Nov 2019 - 2:00

score 439 · Answer 13 · 2026-03-26 04:36:44

439

Views

Compute $\int_0^1\frac{\ln x\operatorname{Li}_2(x^2)}{\sqrt{1-x^2}}dx$

Published on 26 Mar 2026 - 4:36

score 626 · Answer 14 · 2019-11-08 22:25:03

626

Views

Prove $\int_0^1\frac{x^{2n}}{1+x}dx=\ln2+H_n-H_{2n}$

Published on 08 Nov 2019 - 22:25

score 225 · Answer 15 · 2026-03-26 01:11:55

225

Views

How to prove this series identity $\sum_{n=1}^{\infty}\left(\frac{1}{n}\sum_{k=n+1}^{\infty}\frac{1}{k^3}\right)=\frac{\pi^{4}}{360}$?

Published on 26 Mar 2026 - 1:11

List Question

Calculating logarithmic integrals without using the derivatives of Beta function.

Calulating $\int_0^1\frac{5\ln^4x\ln^4(1-x)-8\ln^5x\ln^3(1-x)+4\ln^2(1-x)\ln^6x}{x(1-x)}\ dx$

A bound of the harmonic series of squares.

Advanced : Compute $\sum_{n=1}^\infty\frac{H_n^4-6H_n^2H_n^{(2)}+8H_nH_n^{(3)}+3\left(H_n^{(2)}\right)^2-6H_n^{(4)}}{n^5}$

A group of advanced harmonic series with weight of 9

Is there a closed form for $\int_0^1\frac{\ln^4(1-x)\operatorname{Li}_4(x)}{x}dx\ ?$

Positivity of a certain sum of Stirling numbers

Closed form for $\int_{0}^1 \frac{\ln(x)\ln(1-x)^b}{x^2}dx$

Generalization of Harmonic series and Euler-Mascheroni Constant

Closed Form for $\int_{0}^1\frac{\ln(x)\ln^a(1-x)}{x^3}dx$

Finding the sum of $\sum^\infty_{k=1}{\frac{1}{k(k+2)}}$ with partial fraction decomposition

Prove $\zeta(3)=2\sum_{n=1}^\infty\frac{H_n}{n}\left[\frac1{4^n}{2n\choose n}\left(H_{2n}-H_n-\frac1{2n}-\ln2\right)+\frac1{2n}\right]$

Compute $\int_0^1\frac{\ln x\operatorname{Li}_2(x^2)}{\sqrt{1-x^2}}dx$

Prove $\int_0^1\frac{x^{2n}}{1+x}dx=\ln2+H_n-H_{2n}$

How to prove this series identity $\sum_{n=1}^{\infty}\left(\frac{1}{n}\sum_{k=n+1}^{\infty}\frac{1}{k^3}\right)=\frac{\pi^{4}}{360}$?

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