Question

Proof of $\sum_{n\ge0}^{ }n^{m}z^{n}=\sum_{j=0}^{m}{m+1\brace j+1}\frac{\left(-1\right)^{m-j}j!}{\left(1-z\right)^{j+1}}$

score 107 · Answer 1 · 2026-03-27 04:23:49

107

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Proof of $\sum_{n\ge0}^{ }n^{m}z^{n}=\sum_{j=0}^{m}{m+1\brace j+1}\frac{\left(-1\right)^{m-j}j!}{\left(1-z\right)^{j+1}}$

Published on 27 Mar 2026 - 4:23

score 521 · Answer 2 · 2026-03-27 04:21:59

521

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Range of $m,n$ in $\sum_{j=0}^{m}{m\brace j}\binom{n}{j}j!=n^{m}$

Published on 27 Mar 2026 - 4:21

score 340 · Answer 3 · 2026-04-14 13:48:21

340

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Cauchy product summation converges

Published on 14 Apr 2026 - 13:48

score 159 · Answer 4 · 2026-04-15 04:59:04

159

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How is $n\sum^n_{j=1}(x_j-\overline{x})^2$ derived?

Published on 15 Apr 2026 - 4:59

score 26 · Answer 5 · 2026-04-16 05:49:00

26

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Proving that Cauchy Summation converges

Published on 16 Apr 2026 - 5:49

score 86 · Answer 6 · 2026-04-12 18:34:15

86

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Sum of product of binomials

Published on 12 Apr 2026 - 18:34

score 391 · Answer 7 · 2026-04-12 12:38:46

391

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Dini's theorem (specific case)

Published on 12 Apr 2026 - 12:38

score 62 · Answer 8 · 2026-04-14 09:40:21

62

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For which values the following summation converges

Published on 14 Apr 2026 - 9:40

score 130 · Answer 9 · 2026-04-14 07:02:11

130

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Show that $\binom{n-1}{r-1}=\sum^{r-1}_{k=0}(-1)^k\binom rk\binom{n+r-k-1}{r-k-1}$

Published on 14 Apr 2026 - 7:02

score 145 · Answer 10 · 2026-04-14 01:10:23

145

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sum of the first $n$ terms of a quadratic sequence

Published on 14 Apr 2026 - 1:10

score 95 · Answer 11 · 2026-04-17 00:44:38

95

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Simplify $\sum_{k = 1}^{\left\lfloor{N/2}\right\rfloor} \sum_{i = 1}^{k} \sum_{d_1, d_2 = 1}^{N} \left[{{d}_{1}\, {d}_{2} = i (2\, k - i)}\right]$

Published on 17 Apr 2026 - 0:44

score 50 · Answer 12 · 2026-04-15 13:40:01

50

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Prove combinatorially: $\sum_{r=0}^n (-1)^r \binom{n}{r}CC^{k-r}_n = 0$

Published on 15 Apr 2026 - 13:40

score 369 · Answer 13 · 2026-04-17 08:09:54

369

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Express $\sum_{i=0}^n (3^3 − 6 + 2)$ as a polynomial $p(n)$

Published on 17 Apr 2026 - 8:09

score 40 · Answer 14 · 2026-05-05 17:43:37

40

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Why does $\sum_{i=1}^n a = (n+1)a$

Published on 05 May 2026 - 17:43

score 83 · Answer 15 · 2026-04-14 17:06:27

83

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Why is $\prod_{n}^{n-1} = 1$ and $\sum_{k=j+1}^{j}=0$?

Published on 14 Apr 2026 - 17:06

List Question

Proof of $\sum_{n\ge0}^{ }n^{m}z^{n}=\sum_{j=0}^{m}{m+1\brace j+1}\frac{\left(-1\right)^{m-j}j!}{\left(1-z\right)^{j+1}}$

Range of $m,n$ in $\sum_{j=0}^{m}{m\brace j}\binom{n}{j}j!=n^{m}$

Cauchy product summation converges

How is $n\sum^n_{j=1}(x_j-\overline{x})^2$ derived?

Proving that Cauchy Summation converges

Sum of product of binomials

Dini's theorem (specific case)

For which values the following summation converges

Show that $\binom{n-1}{r-1}=\sum^{r-1}_{k=0}(-1)^k\binom rk\binom{n+r-k-1}{r-k-1}$

sum of the first $n$ terms of a quadratic sequence

Simplify $\sum_{k = 1}^{\left\lfloor{N/2}\right\rfloor} \sum_{i = 1}^{k} \sum_{d_1, d_2 = 1}^{N} \left[{{d}_{1}\, {d}_{2} = i (2\, k - i)}\right]$

Prove combinatorially: $\sum_{r=0}^n (-1)^r \binom{n}{r}CC^{k-r}_n = 0$

Express $\sum_{i=0}^n (3^3 − 6 + 2)$ as a polynomial $p(n)$

Why does $\sum_{i=1}^n a = (n+1)a$

Why is $\prod_{n}^{n-1} = 1$ and $\sum_{k=j+1}^{j}=0$?

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