Question

Proving $\sum_{n=-\infty}^\infty n^2e^{-\pi n^2}=\frac{\Gamma (1/4)}{4\sqrt{2}\pi^{7/4}}$

score 173 · Answer 1 · 2026-03-26 17:32:46

173

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Proving $\sum_{n=-\infty}^\infty n^2e^{-\pi n^2}=\frac{\Gamma (1/4)}{4\sqrt{2}\pi^{7/4}}$

Published on 26 Mar 2026 - 17:32

score 52 · Answer 2 · 2026-03-27 23:14:24

52

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Does the rest of this family of continued fractions have closed forms?

Published on 27 Mar 2026 - 23:14

score 186 · Answer 3 · 2026-03-29 03:51:13

186

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The closed-form of $1-5\left(\frac{1}{2}\right)^k+9\left(\frac{1\cdot3}{2\cdot4}\right)^k-13\left(\frac{1\cdot3\cdot5}{2\cdot4\cdot6}\right)^k+\dots$?

Published on 29 Mar 2026 - 3:51

score 198 · Answer 4 · 2023-10-28 13:18:37

198

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On $\int_0^1\prod_{k=1}^\infty (1-x^k)\, dx$

Published on 28 Oct 2023 - 13:18

score 362 · Answer 5 · 2026-03-30 11:40:08

362

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Prove $\,_3F_2\left ( \frac12,\frac12,1;\frac34,\frac34;-\frac18 \right ) =\frac23+\frac{\Gamma\left ( \frac34 \right )^2}{3\sqrt{\pi} }$

Published on 30 Mar 2026 - 11:40

score 127 · Answer 6 · 2023-10-30 17:04:43

127

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Closed Formula for Double Summation

Published on 30 Oct 2023 - 17:04

score 75 · Answer 7 · 2026-03-29 03:45:07

75

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Why $1+\frac{1}{3}-\frac{1}{5}-\frac{1}{7}+\cdots=\frac{\pi}{2\sqrt{2}}$?

Published on 29 Mar 2026 - 3:45

score 264 · Answer 8 · 2026-03-26 06:31:55

264

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Using Ramanujan-type series for $1/\pi^m$ to find formulas for $\zeta(2),\, \zeta(3),\, \zeta(4)$?

Published on 26 Mar 2026 - 6:31

score 83 · Answer 9 · 2023-11-02 16:41:13

83

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KL divergence between binomial distribution and Poisson distribution

Published on 02 Nov 2023 - 16:41

score 66 · Answer 10 · 2023-11-03 08:39:57

66

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Does $\sum_{n=1}^{\infty} nx^{2^n}$ has a closed form when $0<x<1$?

Published on 03 Nov 2023 - 8:39

score 78 · Answer 11 · 2023-11-03 17:55:06

78

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What is $\sum_{n=-\infty}^\infty \frac{1}{(z-n)^\alpha}$?

Published on 03 Nov 2023 - 17:55

score 156 · Answer 12 · 2023-11-14 11:31:50

156

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Closed form: $\displaystyle\int_0^\infty \! \prod_{i=1}^n\frac{1}{a_i^2 + x^2} \, dx$

Published on 14 Nov 2023 - 11:31

score 144 · Answer 13 · 2026-03-25 23:18:50

144

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Simplifying a complicated transcendental equation

Published on 25 Mar 2026 - 23:18

score 136 · Answer 14 · 2023-11-15 09:16:11

136

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Can an exact vector field have loops as solutions?

Published on 15 Nov 2023 - 9:16

score 161 · Answer 15 · 2023-11-16 10:11:14

161

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Oxford interview question was supposedly $\int\frac{1}{1-\ln x}dx$, but that's impossible. Maybe they meant $\int_a^b \frac{1}{1-\ln x}dx$?

Published on 16 Nov 2023 - 10:11

List Question

Proving $\sum_{n=-\infty}^\infty n^2e^{-\pi n^2}=\frac{\Gamma (1/4)}{4\sqrt{2}\pi^{7/4}}$

Does the rest of this family of continued fractions have closed forms?

The closed-form of $1-5\left(\frac{1}{2}\right)^k+9\left(\frac{1\cdot3}{2\cdot4}\right)^k-13\left(\frac{1\cdot3\cdot5}{2\cdot4\cdot6}\right)^k+\dots$?

On $\int_0^1\prod_{k=1}^\infty (1-x^k)\, dx$

Prove $\,_3F_2\left ( \frac12,\frac12,1;\frac34,\frac34;-\frac18 \right ) =\frac23+\frac{\Gamma\left ( \frac34 \right )^2}{3\sqrt{\pi} }$

Closed Formula for Double Summation

Why $1+\frac{1}{3}-\frac{1}{5}-\frac{1}{7}+\cdots=\frac{\pi}{2\sqrt{2}}$?

Using Ramanujan-type series for $1/\pi^m$ to find formulas for $\zeta(2),\, \zeta(3),\, \zeta(4)$?

KL divergence between binomial distribution and Poisson distribution

Does $\sum_{n=1}^{\infty} nx^{2^n}$ has a closed form when $0<x<1$?

What is $\sum_{n=-\infty}^\infty \frac{1}{(z-n)^\alpha}$?

Closed form: $\displaystyle\int_0^\infty \! \prod_{i=1}^n\frac{1}{a_i^2 + x^2} \, dx$

Simplifying a complicated transcendental equation

Can an exact vector field have loops as solutions?

Oxford interview question was supposedly $\int\frac{1}{1-\ln x}dx$, but that's impossible. Maybe they meant $\int_a^b \frac{1}{1-\ln x}dx$?

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