Question

How to prove that $\sum _{k=1} ^N \frac 1 {1-\cos \frac {k\pi} N} = \frac {2N^2 + 1} 6$?

score 922 · Answer 1 · 2016-09-21 08:21:29

922

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How to prove that $\sum _{k=1} ^N \frac 1 {1-\cos \frac {k\pi} N} = \frac {2N^2 + 1} 6$?

Published on 21 Sep 2016 - 8:21

score 579 · Answer 2 · 2026-03-27 17:04:21

579

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Conjecture $\sum_{n=1}^\infty\frac{n^2}{(-1)^n \cosh(\pi n\sqrt{3})-1}=\frac{1}{12\pi^2}$

Published on 27 Mar 2026 - 17:04

score 526 · Answer 3 · 2016-09-24 04:25:48

526

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Integrating the log logistic density

Published on 24 Sep 2016 - 4:25

score 1k · Answer 4 · 2016-09-26 01:23:56

1k

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Evaluation of Half Dirac Delta Function

Published on 26 Sep 2016 - 1:23

score 272 · Answer 5 · 2016-09-26 14:13:24

272

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Find $\sum\limits_{k=1}^n \frac{1}{k}\binom{n}{k}$

Published on 26 Sep 2016 - 14:13

score 104 · Answer 6 · 2016-09-26 15:02:08

104

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Closed form solution for infinite summation

Published on 26 Sep 2016 - 15:02

score 4.4k · Answer 7 · 2026-03-31 13:43:04

4.4k

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Finding a closed form for a recurrence relation $a_n=3a_{n-1}+4a_{n-2}$

Published on 31 Mar 2026 - 13:43

score 1.3k · Answer 8 · 2026-03-26 12:37:19

1.3k

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On the general form of the family $\sum_{n=1}^\infty \frac{n^{k}}{e^{2n\pi}-1} $

Published on 26 Mar 2026 - 12:37

score 104 · Answer 9 · 2016-09-28 18:35:35

104

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Proving the formula of the geometric sum through induction

Published on 28 Sep 2016 - 18:35

score 228 · Answer 10 · 2016-09-29 04:56:50

228

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How to show $e^{-\sqrt{2\lambda }x}=\int_0^{\infty} e^{-\lambda u} (2\pi u^3)^{-1/2} x \exp(-\frac{x^2}{2u}) du$?

Published on 29 Sep 2016 - 4:56

score 355 · Answer 11 · 2016-09-29 18:53:53

355

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Solving $\int_0^\infty\dfrac{dx}{(1+x^n)^n}=1$

Published on 29 Sep 2016 - 18:53

score 1.2k · Answer 12 · 2016-09-30 04:35:22

1.2k

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How to produce the closed form of finite summations of the form $\sum_{i=0}^n ia^i$.

Published on 30 Sep 2016 - 4:35

score 57 · Answer 13 · 2016-09-30 08:01:36

57

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Show that $ f(t) = \begin{cases} e^{-1/t} & x > 0 \\ 0 & x \le 0 \end{cases}$ is in $C^{\infty}$.

Published on 30 Sep 2016 - 8:01

score 144 · Answer 14 · 2026-03-31 13:32:02

144

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2 conjectured recursion limits for $e$ and $\pi$.

Published on 31 Mar 2026 - 13:32

score 195 · Answer 15 · 2016-10-01 17:11:51

195

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Mellin transform of $\frac{\sinh (\pi x)}{(\cosh (\pi x)-1)^3}$

Published on 01 Oct 2016 - 17:11

List Question

How to prove that $\sum _{k=1} ^N \frac 1 {1-\cos \frac {k\pi} N} = \frac {2N^2 + 1} 6$?

Conjecture $\sum_{n=1}^\infty\frac{n^2}{(-1)^n \cosh(\pi n\sqrt{3})-1}=\frac{1}{12\pi^2}$

Integrating the log logistic density

Evaluation of Half Dirac Delta Function

Find $\sum\limits_{k=1}^n \frac{1}{k}\binom{n}{k}$

Closed form solution for infinite summation

Finding a closed form for a recurrence relation $a_n=3a_{n-1}+4a_{n-2}$

On the general form of the family $\sum_{n=1}^\infty \frac{n^{k}}{e^{2n\pi}-1} $

Proving the formula of the geometric sum through induction

How to show $e^{-\sqrt{2\lambda }x}=\int_0^{\infty} e^{-\lambda u} (2\pi u^3)^{-1/2} x \exp(-\frac{x^2}{2u}) du$?

Solving $\int_0^\infty\dfrac{dx}{(1+x^n)^n}=1$

How to produce the closed form of finite summations of the form $\sum_{i=0}^n ia^i$.

Show that $ f(t) = \begin{cases} e^{-1/t} & x > 0 \\ 0 & x \le 0 \end{cases}$ is in $C^{\infty}$.

2 conjectured recursion limits for $e$ and $\pi$.

Mellin transform of $\frac{\sinh (\pi x)}{(\cosh (\pi x)-1)^3}$

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