Question

The best notation for this identity involving pentagonal numbers $\omega(n)$ and the $3x+1$ map

score 142 · Answer 1 · 2026-03-28 23:59:50

142

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The best notation for this identity involving pentagonal numbers $\omega(n)$ and the $3x+1$ map

Published on 28 Mar 2026 - 23:59

score 374 · Answer 2 · 2016-05-29 22:08:53

374

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Conjecture $\sum_{n=1}^\infty\frac{\ln(n+2)}{n\,(n+1)}\,\stackrel{\color{gray}?}=\,{\large\int}_0^1\frac{x\,(\ln x-1)}{\ln(1-x)}\,dx$

Published on 29 May 2016 - 22:08

score 489 · Answer 3 · 2026-04-01 18:50:44

489

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Integral ${\large\int}_0^\infty\big(2J_0(2x)^2+2Y_0(2x)^2-J_0(x)^2-Y_0(x)^2\big)\,dx$

Published on 01 Apr 2026 - 18:50

score 257 · Answer 4 · 2016-05-30 10:18:54

257

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Evaluating $\int_{-1}^{1} \frac{dx}{(e^x+1)(x^2+1)}$

Published on 30 May 2016 - 10:18

score 300 · Answer 5 · 2016-05-30 22:25:37

300

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Proving $\int_{0}^{\infty}\frac{4x^2}{(x^4+2x^2+2)^2}dx\stackrel?=\frac{\pi}{4}\sqrt{5\sqrt2-7}$

Published on 30 May 2016 - 22:25

score 91 · Answer 6 · 2016-06-02 09:07:01

91

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Solving a linear recurrence equation: $S_n=\sum_{i=0}^{n-1}a_{n-i}S_i$

Published on 02 Jun 2016 - 9:07

score 193 · Answer 7 · 2026-03-29 03:35:35

193

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Other types of closed form for $\sum_{n=1}^\infty \frac{\cos(nx)}{n}$ in specific interval

Published on 29 Mar 2026 - 3:35

score 308 · Answer 8 · 2016-06-02 19:03:07

308

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Prove $1-2\int_{0}^{\infty}{\tan^{-1}(x)\over e^{x\pi}-1}dx=\ln(2)$

Published on 02 Jun 2016 - 19:03

score 244 · Answer 9 · 2016-06-03 05:00:31

244

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Exponential series property: $\alpha (z)=\sum_{n=-\infty }^{\infty }e^{-zn^{2}\pi}$

Published on 03 Jun 2016 - 5:00

score 359 · Answer 10 · 2016-06-03 14:45:10

359

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Evaluation of $\sum^{\infty}_{n=0}\frac{1}{16^n}\binom{2n}{n}.$

Published on 03 Jun 2016 - 14:45

score 1.9k · Answer 11 · 2026-03-26 08:00:32

1.9k

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Family of definite integrals involving Dedekind eta function of a complex argument, $\int_0^{\infty} \eta^k(ix)dx$

Published on 26 Mar 2026 - 8:00

score 367 · Answer 12 · 2016-06-04 18:31:59

367

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Prove $\int_{0}^{2\pi}{x\sin^3(x)\over 1+\cos^2(x)}dx=2\pi-\pi^2$

Published on 04 Jun 2016 - 18:31

score 2.8k · Answer 13 · 2026-04-02 08:02:17

2.8k

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Summation Closed form for floor$\left(\log_n\right)$

Published on 02 Apr 2026 - 8:02

score 487 · Answer 14 · 2016-06-06 21:58:12

487

Views

General Two-State Markov Chain: $P(X_{n}=1)=\frac{b}{a+b}+(1-a-b)^n \big(P(X_0=1)-\frac{b}{a+b}\big)$

Published on 06 Jun 2016 - 21:58

score 160 · Answer 15 · 2016-06-08 10:33:19

160

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Evaluate $\int_0^{\pi/2}(\sin x)^n e^{-(2+\cos x)\log k}dx$ for fixed integers $n,k\geq 1$

Published on 08 Jun 2016 - 10:33

List Question

The best notation for this identity involving pentagonal numbers $\omega(n)$ and the $3x+1$ map

Conjecture $\sum_{n=1}^\infty\frac{\ln(n+2)}{n\,(n+1)}\,\stackrel{\color{gray}?}=\,{\large\int}_0^1\frac{x\,(\ln x-1)}{\ln(1-x)}\,dx$

Integral ${\large\int}_0^\infty\big(2J_0(2x)^2+2Y_0(2x)^2-J_0(x)^2-Y_0(x)^2\big)\,dx$

Evaluating $\int_{-1}^{1} \frac{dx}{(e^x+1)(x^2+1)}$

Proving $\int_{0}^{\infty}\frac{4x^2}{(x^4+2x^2+2)^2}dx\stackrel?=\frac{\pi}{4}\sqrt{5\sqrt2-7}$

Solving a linear recurrence equation: $S_n=\sum_{i=0}^{n-1}a_{n-i}S_i$

Other types of closed form for $\sum_{n=1}^\infty \frac{\cos(nx)}{n}$ in specific interval

Prove $1-2\int_{0}^{\infty}{\tan^{-1}(x)\over e^{x\pi}-1}dx=\ln(2)$

Exponential series property: $\alpha (z)=\sum_{n=-\infty }^{\infty }e^{-zn^{2}\pi}$

Evaluation of $\sum^{\infty}_{n=0}\frac{1}{16^n}\binom{2n}{n}.$

Family of definite integrals involving Dedekind eta function of a complex argument, $\int_0^{\infty} \eta^k(ix)dx$

Prove $\int_{0}^{2\pi}{x\sin^3(x)\over 1+\cos^2(x)}dx=2\pi-\pi^2$

Summation Closed form for floor$\left(\log_n\right)$

General Two-State Markov Chain: $P(X_{n}=1)=\frac{b}{a+b}+(1-a-b)^n \big(P(X_0=1)-\frac{b}{a+b}\big)$

Evaluate $\int_0^{\pi/2}(\sin x)^n e^{-(2+\cos x)\log k}dx$ for fixed integers $n,k\geq 1$

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