Question

Prove $\sum_{k=0}^\infty \frac{(-1)^k}{3k+2} = \frac{1}{9}\left(\sqrt{3}\pi-3\ln\,2\right)$

score 101 · Answer 1 · 2026-04-15 22:03:55

101

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Prove $\sum_{k=0}^\infty \frac{(-1)^k}{3k+2} = \frac{1}{9}\left(\sqrt{3}\pi-3\ln\,2\right)$

Published on 15 Apr 2026 - 22:03

score 84 · Answer 2 · 2026-04-13 00:25:41

84

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Bound on $\sum_{\mathbb{Z}^d}|\widehat{f}({\boldsymbol{k}})||{\boldsymbol{k}}|^7$.

Published on 13 Apr 2026 - 0:25

score 56 · Answer 3 · 2026-04-14 17:43:17

56

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limit of a recursive sequence_1

Published on 14 Apr 2026 - 17:43

score 108 · Answer 4 · 2026-04-13 20:14:27

108

Views

Finding the summation of these two Infinite Series

Published on 13 Apr 2026 - 20:14

score 158 · Answer 5 · 2026-04-15 23:25:04

158

Views

If we have $\gcd(a_n,a_m)=a_{\gcd(n,m)}$ for each pair then $a_1,a_2,...$ is Fibonacci sequence?

Published on 15 Apr 2026 - 23:25

score 64 · Answer 6 · 2026-04-14 06:38:48

64

Views

Prove that the series with terms $a_n$ diverges.

Published on 14 Apr 2026 - 6:38

score 1.2k · Answer 7 · 2026-04-20 19:17:12

1.2k

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Some details about 'Collatz Conjecture'?

Published on 20 Apr 2026 - 19:17

score 181 · Answer 8 · 2026-04-21 04:10:49

181

Views

Can you write something like $3\cdot5\cdot7\cdots(2n+1)$ as $(2n+1)!$?

Published on 21 Apr 2026 - 4:10

score 323 · Answer 9 · 2026-04-15 13:48:05

323

Views

Convergence/Divergence of the recursive series

Published on 15 Apr 2026 - 13:48

score 49 · Answer 10 · 2026-04-16 23:41:54

49

Views

The proof that if $a_{1}>1$ and $a_{n+1}=2-\frac{1}{a_{n}}$ for $n\geq 1$, then $\{a_{n}\}$ is monotone

Published on 16 Apr 2026 - 23:41

score 326 · Answer 11 · 2026-04-17 14:07:10

326

Views

If $\,a_n\searrow 0\,$ and $\,\sum_{n=1}^\infty a_n<\infty,\,$ does this imply that $\,n\log n\, a_n\to 0$?

Published on 17 Apr 2026 - 14:07

score 294 · Answer 12 · 2026-04-21 04:22:16

294

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How to find the sum of: $a^2+(a+d)^2+(a+2d)^2+(a+3d)^2+\cdots+[a+(n-1)d]^2?$

Published on 21 Apr 2026 - 4:22

score 50 · Answer 13 · 2026-04-14 10:53:21

50

Views

Irrationality based on series expansion condition

Published on 14 Apr 2026 - 10:53

score 8.4k · Answer 14 · 2026-04-14 09:35:51

8.4k

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Can the product of infinitely many elements from $\mathbb Q$ be irrational?

Published on 14 Apr 2026 - 9:35

score 108 · Answer 15 · 2026-04-16 00:07:29

108

Views

How do I show that $h_n=\begin{cases}n!2^{\frac{n}{2}-1} , \text {if $n$ is even} \\ 0, \text{if $n$ is odd} \end{cases}$

Published on 16 Apr 2026 - 0:07

List Question

Prove $\sum_{k=0}^\infty \frac{(-1)^k}{3k+2} = \frac{1}{9}\left(\sqrt{3}\pi-3\ln\,2\right)$

Bound on $\sum_{\mathbb{Z}^d}|\widehat{f}({\boldsymbol{k}})||{\boldsymbol{k}}|^7$.

limit of a recursive sequence_1

Finding the summation of these two Infinite Series

If we have $\gcd(a_n,a_m)=a_{\gcd(n,m)}$ for each pair then $a_1,a_2,...$ is Fibonacci sequence?

Prove that the series with terms $a_n$ diverges.

Some details about 'Collatz Conjecture'?

Can you write something like $3\cdot5\cdot7\cdots(2n+1)$ as $(2n+1)!$?

Convergence/Divergence of the recursive series

The proof that if $a_{1}>1$ and $a_{n+1}=2-\frac{1}{a_{n}}$ for $n\geq 1$, then $\{a_{n}\}$ is monotone

If $\,a_n\searrow 0\,$ and $\,\sum_{n=1}^\infty a_n<\infty,\,$ does this imply that $\,n\log n\, a_n\to 0$?

How to find the sum of: $a^2+(a+d)^2+(a+2d)^2+(a+3d)^2+\cdots+[a+(n-1)d]^2?$

Irrationality based on series expansion condition

Can the product of infinitely many elements from $\mathbb Q$ be irrational?

How do I show that $h_n=\begin{cases}n!2^{\frac{n}{2}-1} , \text {if $n$ is even} \\ 0, \text{if $n$ is odd} \end{cases}$

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