Question

Is there a simple expression for $\sum_{n=0}^{\infty} \left[ (4x)^n \frac{(n!)^2}{(2n+1)!} \right]^2?$

score 258 · Answer 1 · 2014-10-28 13:04:08

258

Views

Is there a simple expression for $\sum_{n=0}^{\infty} \left[ (4x)^n \frac{(n!)^2}{(2n+1)!} \right]^2?$

Published on 28 Oct 2014 - 13:04

score 182 · Answer 2 · 2014-10-28 17:42:02

182

Views

Closed form of a sum

Published on 28 Oct 2014 - 17:42

score 450 · Answer 3 · 2014-10-28 19:10:18

450

Views

Evaluation of $\int_0^\pi \! \ln\left(1-2\alpha\cos x+\alpha^2\right) \mathrm{d}x$

Published on 28 Oct 2014 - 19:10

score 142 · Answer 4 · 2014-10-29 12:19:54

142

Views

using complex or real analysis solve $\int_{0}^{\pi/2}\frac{x^m}{\sin x}dx$

Published on 29 Oct 2014 - 12:19

score 388 · Answer 5 · 2014-10-30 14:12:06

388

Views

Evaluating by real methods $\int_0^{\pi/2} \frac{x^5}{2-\cos^2(x)}\ dx$

Published on 30 Oct 2014 - 14:12

score 479 · Answer 6 · 2014-10-31 16:54:17

479

Views

Evaluating $\sum_{n=1}^{\infty} (-1)^{n-1}\frac{H_{2n}}{n}$

Published on 31 Oct 2014 - 16:54

score 329 · Answer 7 · 2026-03-26 02:47:39

329

Views

How to prove $\int_{0}^{-1} \frac{\operatorname{Li}_2(x)}{(1-x)^2} dx=\frac{\pi^2}{24}-\frac{\ln^2(2)}{2} $

Published on 26 Mar 2026 - 2:47

score 269 · Answer 8 · 2026-03-28 11:34:14

269

Views

Integral of a product of five Bessel functions of order $0$

Published on 28 Mar 2026 - 11:34

score 883 · Answer 9 · 2014-10-31 23:26:03

883

Views

An integral by O. Furdui $\int_0^1 \log^2(\sqrt{1+x}-\sqrt{1-x}) \ dx$

Published on 31 Oct 2014 - 23:26

score 692 · Answer 10 · 2014-11-01 11:50:03

692

Views

Closed form: $\int_0^\pi \left( \frac{2 + 2\cos(x) -\cos((k-\frac{1}{2})x) -2\cos((k+\frac{1}{2})x) - cos((k+\frac{3}{2})x)}{1-\cos(2x)}\right)dx $

Published on 01 Nov 2014 - 11:50

score 306 · Answer 11 · 2014-11-01 22:57:05

306

Views

Closed form of $\displaystyle\int_{0}^{\pi/4}\int_{\pi/2}^{\pi}\frac{(\cos x-\sin x)^{y-2}}{(\cos x+\sin x)^{y+2}}\, dy\, dx$

Published on 01 Nov 2014 - 22:57

score 174 · Answer 12 · 2014-11-02 07:30:00

174

Views

what is the summation of such a finite sequence?

Published on 02 Nov 2014 - 7:30

score 2k · Answer 13 · 2014-11-02 07:47:36

2k

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Evaluate $ \int_{0}^{\pi/2}\frac{1+\tanh x}{1+\tan x}dx $

Published on 02 Nov 2014 - 7:47

score 176 · Answer 14 · 2014-11-02 07:51:30

176

Views

summation of a finite sequence?

Published on 02 Nov 2014 - 7:51

score 443 · Answer 15 · 2026-03-27 18:57:08

443

Views

The Integral of Multiple Tangent Functions

Published on 27 Mar 2026 - 18:57

List Question

Is there a simple expression for $\sum_{n=0}^{\infty} \left[ (4x)^n \frac{(n!)^2}{(2n+1)!} \right]^2?$

Closed form of a sum

Evaluation of $\int_0^\pi \! \ln\left(1-2\alpha\cos x+\alpha^2\right) \mathrm{d}x$

using complex or real analysis solve $\int_{0}^{\pi/2}\frac{x^m}{\sin x}dx$

Evaluating by real methods $\int_0^{\pi/2} \frac{x^5}{2-\cos^2(x)}\ dx$

Evaluating $\sum_{n=1}^{\infty} (-1)^{n-1}\frac{H_{2n}}{n}$

How to prove $\int_{0}^{-1} \frac{\operatorname{Li}_2(x)}{(1-x)^2} dx=\frac{\pi^2}{24}-\frac{\ln^2(2)}{2} $

Integral of a product of five Bessel functions of order $0$

An integral by O. Furdui $\int_0^1 \log^2(\sqrt{1+x}-\sqrt{1-x}) \ dx$

Closed form: $\int_0^\pi \left( \frac{2 + 2\cos(x) -\cos((k-\frac{1}{2})x) -2\cos((k+\frac{1}{2})x) - cos((k+\frac{3}{2})x)}{1-\cos(2x)}\right)dx $

Closed form of $\displaystyle\int_{0}^{\pi/4}\int_{\pi/2}^{\pi}\frac{(\cos x-\sin x)^{y-2}}{(\cos x+\sin x)^{y+2}}\, dy\, dx$

what is the summation of such a finite sequence?

Evaluate $ \int_{0}^{\pi/2}\frac{1+\tanh x}{1+\tan x}dx $

summation of a finite sequence?

The Integral of Multiple Tangent Functions

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