Question

For what $d$ does $\sum\limits_{m=-\infty}^{\infty}\sum\limits_{n=0}^{\infty}\frac{1}{2^{d(2n+|m|)}}\binom{2n+|m|}{n}^d$ converge?

score 81 · Answer 1 · 2017-01-13 07:37:00

81

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For what $d$ does $\sum\limits_{m=-\infty}^{\infty}\sum\limits_{n=0}^{\infty}\frac{1}{2^{d(2n+|m|)}}\binom{2n+|m|}{n}^d$ converge?

Published on 13 Jan 2017 - 7:37

score 46 · Answer 2 · 2026-04-11 15:06:42

46

Views

count integral solutions to equation using probability

Published on 11 Apr 2026 - 15:06

score 86 · Answer 3 · 2026-04-11 16:07:13

86

Views

Prove trigonometric identity using combinatorics

Published on 11 Apr 2026 - 16:07

score 742 · Answer 4 · 2017-01-14 22:15:38

742

Views

$\sum_{1\le k\le n,~k\mathrm{~odd}}k\binom{n}{k}=n\cdot2^{n-2}$ for $n>1$

Published on 14 Jan 2017 - 22:15

score 127 · Answer 5 · 2026-04-12 16:37:10

127

Views

Sum of series involving binomial coefficients

Published on 12 Apr 2026 - 16:37

score 871 · Answer 6 · 2026-04-10 15:27:42

871

Views

Coefficient in binomial expansion for negative terms

Published on 10 Apr 2026 - 15:27

score 293 · Answer 7 · 2017-01-16 11:52:19

293

Views

Evaluate $S_n(n)$ for $S_r(n)=\sum_{m=0}^n(-1)^m m^r\binom{n}{m}$

Published on 16 Jan 2017 - 11:52

score 71 · Answer 8 · 2026-04-13 12:07:49

71

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Calculate $\sum\limits^\infty_{n=0}\sum\limits^n_{k=0} \binom n k(-\frac 1 3)^k$?

Published on 13 Apr 2026 - 12:07

score 3.9k · Answer 9 · 2017-01-17 21:36:52

3.9k

Views

Picking zero out of zero elements

Published on 17 Jan 2017 - 21:36

score 220 · Answer 10 · 2026-03-31 21:51:47

220

Views

Why is $\sum^\infty_{n=0}\sum^n_{k=0}\frac{b^k}{k!}\cdot\frac{a^{n-k}}{(n-k)!}=\sum^\infty_{n=0}\frac{(a+b)^n}{n!}$?

Published on 31 Mar 2026 - 21:51

score 93 · Answer 11 · 2017-01-18 11:17:41

93

Views

Evaluate: $C_0+\frac{C_1}2+\frac{C_2}3+\cdots\frac{C_n}{n+1}$

Published on 18 Jan 2017 - 11:17

score 238 · Answer 12 · 2026-03-31 21:53:12

238

Views

Prove that $2 < f(n) < 3$ if $n$ is a natural number and $n \ge 2$ and $f(n)=\left (1+\frac{1}{n}\right)^n$

Published on 31 Mar 2026 - 21:53

score 2.3k · Answer 13 · 2026-04-06 22:46:28

2.3k

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Show that if $p$ is a prime satisfying $n<p<2n$ then $\binom{2n}{n}\equiv 0 \pmod p$.

Published on 06 Apr 2026 - 22:46

score 98 · Answer 14 · 2026-04-12 18:05:32

98

Views

What is the result of $\sum\limits_{0}^{n}\binom{n+r}{r}$

Published on 12 Apr 2026 - 18:05

score 210 · Answer 15 · 2026-04-02 11:39:11

210

Views

Why is $\sum_{n\ge1} \binom{s}{n}\left(1 - \zeta(n-s)\right)=2^s$ for all $s \in \mathbb{C}$?

Published on 02 Apr 2026 - 11:39

List Question

For what $d$ does $\sum\limits_{m=-\infty}^{\infty}\sum\limits_{n=0}^{\infty}\frac{1}{2^{d(2n+|m|)}}\binom{2n+|m|}{n}^d$ converge?

count integral solutions to equation using probability

Prove trigonometric identity using combinatorics

$\sum_{1\le k\le n,~k\mathrm{~odd}}k\binom{n}{k}=n\cdot2^{n-2}$ for $n>1$

Sum of series involving binomial coefficients

Coefficient in binomial expansion for negative terms

Evaluate $S_n(n)$ for $S_r(n)=\sum_{m=0}^n(-1)^m m^r\binom{n}{m}$

Calculate $\sum\limits^\infty_{n=0}\sum\limits^n_{k=0} \binom n k(-\frac 1 3)^k$?

Picking zero out of zero elements

Why is $\sum^\infty_{n=0}\sum^n_{k=0}\frac{b^k}{k!}\cdot\frac{a^{n-k}}{(n-k)!}=\sum^\infty_{n=0}\frac{(a+b)^n}{n!}$?

Evaluate: $C_0+\frac{C_1}2+\frac{C_2}3+\cdots\frac{C_n}{n+1}$

Prove that $2 < f(n) < 3$ if $n$ is a natural number and $n \ge 2$ and $f(n)=\left (1+\frac{1}{n}\right)^n$

Show that if $p$ is a prime satisfying $n<p<2n$ then $\binom{2n}{n}\equiv 0 \pmod p$.

What is the result of $\sum\limits_{0}^{n}\binom{n+r}{r}$

Why is $\sum_{n\ge1} \binom{s}{n}\left(1 - \zeta(n-s)\right)=2^s$ for all $s \in \mathbb{C}$?

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