Question

Evaluation of $\sum^{\infty}_{n=0}\frac{999!\cdot n!}{(1001+n)!}$

score 76 · Answer 1 · 2026-04-16 03:18:59

76

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Evaluation of $\sum^{\infty}_{n=0}\frac{999!\cdot n!}{(1001+n)!}$

Published on 16 Apr 2026 - 3:18

score 295 · Answer 2 · 2026-04-16 22:40:16

295

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Determine the general term of the sequence $(a_n)_{n\ge1}$, strictly decreasing

Published on 16 Apr 2026 - 22:40

score 137 · Answer 3 · 2026-04-16 02:19:43

137

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How to represent $x^n$ as a sum of $P_k:= (x)(x-1)\dots(x-k+1)$?

Published on 16 Apr 2026 - 2:19

score 47 · Answer 4 · 2026-04-14 17:01:17

47

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Calculate $\displaystyle\sum_{k=0}^n \displaystyle\binom{n}{k} \dfrac{(-1)^{n-k}k^l}{l!}$

Published on 14 Apr 2026 - 17:01

score 52 · Answer 5 · 2026-04-17 12:31:41

52

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Solve the following recurrence relation: $f(N,d) = pf(N-1,d-1) + (1-p)f(N-1,d+1)$ subject to constraints in the body.

Published on 17 Apr 2026 - 12:31

score 79 · Answer 6 · 2026-04-16 22:51:59

79

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find condition on $a(n)$ such that the $\lim_{n\rightarrow\infty}\frac{a(n)\cdot (n-a(n))}{\log\binom{n}{a(n)}}=+\infty$

Published on 16 Apr 2026 - 22:51

score 143 · Answer 7 · 2026-04-14 01:23:32

143

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Prove that $\displaystyle (r+1) \cdot a_{r+1} = (n-r) \cdot a_{r} + (2n+1-r) \cdot a_{r-1}$

Published on 14 Apr 2026 - 1:23

score 59 · Answer 8 · 2026-04-14 16:24:24

59

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Is this formula for number of binary relations correct?

Published on 14 Apr 2026 - 16:24

score 52 · Answer 9 · 2026-04-25 03:05:15

52

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Why is $\sum_{m=0}^{\lfloor xs\rfloor} 2 \binom{s}{m} p^m (1-p)^{s-m} \leq 2\exp{\left(-\frac{2(\lfloor xs\rfloor - sp)^2}{s}\right)}$

Published on 25 Apr 2026 - 3:05

score 307 · Answer 10 · 2026-04-10 22:13:49

307

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Maximizing expected value when distribution is binomial

Published on 10 Apr 2026 - 22:13

score 374 · Answer 11 · 2026-04-14 04:10:06

374

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Show that ${n \choose k}\leq n^k$

Published on 14 Apr 2026 - 4:10

score 256 · Answer 12 · 2026-04-12 20:59:30

256

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Inequality involving binomial coefficients

Published on 12 Apr 2026 - 20:59

score 1.4k · Answer 13 · 2026-04-16 07:14:38

1.4k

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Evaluate $\sum_{k=1}^{n} (2k-1){n \choose k}$ using calculus

Published on 16 Apr 2026 - 7:14

score 362 · Answer 14 · 2026-04-14 06:29:28

362

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Inequality with sum of Binomial coefficients: $\sum\limits^n_{k=1}k \sqrt{\binom nk}\leq\sqrt{2^{n-1}n^3}$

Published on 14 Apr 2026 - 6:29

score 1k · Answer 15 · 2026-04-13 09:44:08

1k

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Using Binomial coefficient to solve a problem with unfair coins

Published on 13 Apr 2026 - 9:44

List Question

Evaluation of $\sum^{\infty}_{n=0}\frac{999!\cdot n!}{(1001+n)!}$

Determine the general term of the sequence $(a_n)_{n\ge1}$, strictly decreasing

How to represent $x^n$ as a sum of $P_k:= (x)(x-1)\dots(x-k+1)$?

Calculate $\displaystyle\sum_{k=0}^n \displaystyle\binom{n}{k} \dfrac{(-1)^{n-k}k^l}{l!}$

Solve the following recurrence relation: $f(N,d) = pf(N-1,d-1) + (1-p)f(N-1,d+1)$ subject to constraints in the body.

find condition on $a(n)$ such that the $\lim_{n\rightarrow\infty}\frac{a(n)\cdot (n-a(n))}{\log\binom{n}{a(n)}}=+\infty$

Prove that $\displaystyle (r+1) \cdot a_{r+1} = (n-r) \cdot a_{r} + (2n+1-r) \cdot a_{r-1}$

Is this formula for number of binary relations correct?

Why is $\sum_{m=0}^{\lfloor xs\rfloor} 2 \binom{s}{m} p^m (1-p)^{s-m} \leq 2\exp{\left(-\frac{2(\lfloor xs\rfloor - sp)^2}{s}\right)}$

Maximizing expected value when distribution is binomial

Show that ${n \choose k}\leq n^k$

Inequality involving binomial coefficients

Evaluate $\sum_{k=1}^{n} (2k-1){n \choose k}$ using calculus

Inequality with sum of Binomial coefficients: $\sum\limits^n_{k=1}k \sqrt{\binom nk}\leq\sqrt{2^{n-1}n^3}$

Using Binomial coefficient to solve a problem with unfair coins

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