Question

$\sum_{j=1}^n \dfrac{(-1)^j\binom{n}{j-1}}{\sum_{1\le k\le j} k}$ closed form

score 40 · Answer 1 · 2026-04-15 20:11:22

40

Views

$\sum_{j=1}^n \dfrac{(-1)^j\binom{n}{j-1}}{\sum_{1\le k\le j} k}$ closed form

Published on 15 Apr 2026 - 20:11

score 2.1k · Answer 2 · 2026-04-12 11:55:50

2.1k

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Finding a closed form for binomial coefficient summation

Published on 12 Apr 2026 - 11:55

score 368 · Answer 3 · 2026-04-10 16:51:35

368

Views

Convergence and sum of series

Published on 10 Apr 2026 - 16:51

score 713 · Answer 4 · 2026-04-15 22:07:04

713

Views

Show that $\sum^{6}_{i=1} a_{i}=\frac{15}{2}$ and $ \sum^{6}_{i=1} a^{2}_{i}=\frac{45}{4} \implies \prod_{i=1}^{6} a_{i} \leq \frac{5}{2}$

Published on 15 Apr 2026 - 22:07

score 90 · Answer 5 · 2026-04-15 07:36:28

90

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Induction on the equality $ \sum_{i=2}^{n+1} \frac{1}{i(i+1)}= \frac{n}{2(n+2)}$

Published on 15 Apr 2026 - 7:36

score 93 · Answer 6 · 2026-04-17 04:06:44

93

Views

If $S(n)=\sum_{i=1}^{n} u_i \in \Bbb Q$ for every $n$, do we have $S(n) = P(n)/Q(n)$ for some $P, Q\in \mathbb{Q}[x]$?

Published on 17 Apr 2026 - 4:06

score 107 · Answer 7 · 2026-04-12 06:22:35

107

Views

How can this sum $\sum_{k=0}^a \binom{n}{k}$ be simplified?

Published on 12 Apr 2026 - 6:22

score 88 · Answer 8 · 2026-05-11 04:23:40

88

Views

What can be said about the sum $f(x) = \sum_{m=1}^{\infty} e^{-x\sqrt{m}}\cos(x\sqrt{m})$ for $x>0$?

Published on 11 May 2026 - 4:23

score 316 · Answer 9 · 2026-04-14 09:04:52

316

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Prove Without Induction: $\sum\limits_{k=2}^{n} \frac{1}{k(k-1)} = 1 - \frac{1}{n}$

Published on 14 Apr 2026 - 9:04

score 179 · Answer 10 · 2026-04-15 14:42:54

179

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Show that $\frac{1}{2}\le \sum\limits_{k=0}^n\frac{1}{n+k}\le1$

Published on 15 Apr 2026 - 14:42

score 408 · Answer 11 · 2026-04-16 06:57:31

408

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A proof of the identity $ \sum_{k = 0}^{n} \frac{(-1)^{k} \binom{n}{k}}{x + k} = \frac{n!}{(x + 0) (x + 1) \cdots (x + n)} $.

Published on 16 Apr 2026 - 6:57

score 49 · Answer 12 · 2026-04-12 04:32:24

49

Views

Using nested multiple summations with a Random function.

Published on 12 Apr 2026 - 4:32

score 76 · Answer 13 · 2026-04-12 12:23:40

76

Views

How to try solving this bizarre inequality?

Published on 12 Apr 2026 - 12:23

score 174 · Answer 14 · 2026-04-15 02:09:50

174

Views

What is the value of $ \sum\limits_{k=0}^{n-1}\binom {n-k-1}{j-1} \binom {r+k}{j+k}$?

Published on 15 Apr 2026 - 2:09

score 773 · Answer 15 · 2026-04-14 05:39:04

773

Views

Prove binomial identity $\sum^n_{i = 0} \binom i k = \binom{n+1} {k+1}$ by counting the lattice paths

Published on 14 Apr 2026 - 5:39

List Question

$\sum_{j=1}^n \dfrac{(-1)^j\binom{n}{j-1}}{\sum_{1\le k\le j} k}$ closed form

Finding a closed form for binomial coefficient summation

Convergence and sum of series

Show that $\sum^{6}_{i=1} a_{i}=\frac{15}{2}$ and $ \sum^{6}_{i=1} a^{2}_{i}=\frac{45}{4} \implies \prod_{i=1}^{6} a_{i} \leq \frac{5}{2}$

Induction on the equality $ \sum_{i=2}^{n+1} \frac{1}{i(i+1)}= \frac{n}{2(n+2)}$

If $S(n)=\sum_{i=1}^{n} u_i \in \Bbb Q$ for every $n$, do we have $S(n) = P(n)/Q(n)$ for some $P, Q\in \mathbb{Q}[x]$?

How can this sum $\sum_{k=0}^a \binom{n}{k}$ be simplified?

What can be said about the sum $f(x) = \sum_{m=1}^{\infty} e^{-x\sqrt{m}}\cos(x\sqrt{m})$ for $x>0$?

Prove Without Induction: $\sum\limits_{k=2}^{n} \frac{1}{k(k-1)} = 1 - \frac{1}{n}$

Show that $\frac{1}{2}\le \sum\limits_{k=0}^n\frac{1}{n+k}\le1$

A proof of the identity $ \sum_{k = 0}^{n} \frac{(-1)^{k} \binom{n}{k}}{x + k} = \frac{n!}{(x + 0) (x + 1) \cdots (x + n)} $.

Using nested multiple summations with a Random function.

How to try solving this bizarre inequality?

What is the value of $ \sum\limits_{k=0}^{n-1}\binom {n-k-1}{j-1} \binom {r+k}{j+k}$?

Prove binomial identity $\sum^n_{i = 0} \binom i k = \binom{n+1} {k+1}$ by counting the lattice paths

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