Question

Summation of binomial-like terms

score 66 · Answer 1 · 2026-04-12 09:33:04

66

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Summation of binomial-like terms

Published on 12 Apr 2026 - 9:33

score 363 · Answer 2 · 2026-03-25 17:20:11

363

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Prove: $\sum\limits_{k=0}^{n}\frac{1}{k+1}\binom{2k}{k}\binom{2n-2k}{n-k} = \binom{2n+1}{n}$

Published on 25 Mar 2026 - 17:20

score 54 · Answer 3 · 2026-04-24 16:05:47

54

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How to parse $\sum_{i=0}^n A_i + B$?

Published on 24 Apr 2026 - 16:05

score 32 · Answer 4 · 2026-04-17 13:57:59

Simplification of $\sum_{\substack{G\in\{0,1\}^n\\G_i=1}} (-1)^{G^\top\vec{1}}\frac{\lambda_i}{G^\top\lambda}.$

score 72 · Answer 5 · 2026-04-09 04:42:19

72

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Expected value for alternating sum

Published on 09 Apr 2026 - 4:42

score 237 · Answer 6 · 2026-05-11 08:02:01

237

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Prove $\sqrt{a^2 + ab + b^2} + \sqrt{b^2 + bc + c^2} + \sqrt{c^2 + ac + a^2} \ge \sqrt{3}(a + b + c)$

Published on 11 May 2026 - 8:02

score 79 · Answer 7 · 2026-04-14 20:02:00

79

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What is $\sum_{k=0}^n {n \choose k} p^k (1-p)^{n-k} e^{-s(k - np)^2}$?

Published on 14 Apr 2026 - 20:02

score 40 · Answer 8 · 2026-04-13 16:23:00

40

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Show that a symmetric matrix $A$ is positive definite given $A_{ij}$

Published on 13 Apr 2026 - 16:23

score 170 · Answer 9 · 2026-04-13 06:38:46

170

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Recurrence relation for the partial sum of an alternating series

Published on 13 Apr 2026 - 6:38

score 187 · Answer 10 · 2026-04-16 12:23:07

187

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Combinatorial proof that $2^{n+1}-1=1+2+4+\dots+2^n$

Published on 16 Apr 2026 - 12:23

score 39 · Answer 11 · 2026-04-16 20:00:39

39

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Prove that $\sum_{i=1}^k i!$ can never be of the form $n^m$ (where $n,m\in\mathbb N$ and $m\ge 2$) for $k\ge 4$.

Published on 16 Apr 2026 - 20:00

score 112 · Answer 12 · 2026-04-24 03:51:54

112

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Evaluate $\int_{0}^{1} \frac{\ln(1-x)\ln^2(1+x)\:dx}{x}$

Published on 24 Apr 2026 - 3:51

score 479 · Answer 13 · 2026-03-26 23:11:35

479

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Change of variables on a double summation to yield a single sum

Published on 26 Mar 2026 - 23:11

score 381 · Answer 14 · 2026-04-11 08:59:39

381

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Comparing summation and integration for non monotonic function

Published on 11 Apr 2026 - 8:59

score 136 · Answer 15 · 2026-04-16 08:16:09

136

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Simplifying $\int_0^a \int_0^b \Bigl(\sum_{m} \sum_{n}\dot{A}_{mn} \sin\frac{m \pi x}{a} \sin\frac{n \pi y}{b}\Bigr)^2dy \,dx$

Published on 16 Apr 2026 - 8:16

List Question

Summation of binomial-like terms

Prove: $\sum\limits_{k=0}^{n}\frac{1}{k+1}\binom{2k}{k}\binom{2n-2k}{n-k} = \binom{2n+1}{n}$

How to parse $\sum_{i=0}^n A_i + B$?

Simplification of $\sum_{\substack{G\in\{0,1\}^n\\G_i=1}} (-1)^{G^\top\vec{1}}\frac{\lambda_i}{G^\top\lambda}.$

Expected value for alternating sum

Prove $\sqrt{a^2 + ab + b^2} + \sqrt{b^2 + bc + c^2} + \sqrt{c^2 + ac + a^2} \ge \sqrt{3}(a + b + c)$

What is $\sum_{k=0}^n {n \choose k} p^k (1-p)^{n-k} e^{-s(k - np)^2}$?

Show that a symmetric matrix $A$ is positive definite given $A_{ij}$

Recurrence relation for the partial sum of an alternating series

Combinatorial proof that $2^{n+1}-1=1+2+4+\dots+2^n$

Prove that $\sum_{i=1}^k i!$ can never be of the form $n^m$ (where $n,m\in\mathbb N$ and $m\ge 2$) for $k\ge 4$.

Evaluate $\int_{0}^{1} \frac{\ln(1-x)\ln^2(1+x)\:dx}{x}$

Change of variables on a double summation to yield a single sum

Comparing summation and integration for non monotonic function

Simplifying $\int_0^a \int_0^b \Bigl(\sum_{m} \sum_{n}\dot{A}_{mn} \sin\frac{m \pi x}{a} \sin\frac{n \pi y}{b}\Bigr)^2dy \,dx$

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