Question

Is there a deep connection between the urn problem and the sum of powers?

score 55 · Answer 1 · 2026-04-08 23:33:33

55

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Is there a deep connection between the urn problem and the sum of powers?

Published on 08 Apr 2026 - 23:33

score 258 · Answer 2 · 2016-12-25 06:59:47

258

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To prove the inequality:- $\frac{4^m}{2\sqrt{m}}\le\binom{2m}{m}\le\frac{4^m}{\sqrt{2m+1}}$

Published on 25 Dec 2016 - 6:59

score 92 · Answer 3 · 2016-12-25 14:22:23

92

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Prove that $\sum _{i=0}^{p} \left( \binom {p+1}{i+1} \times \sum _{j=1}^{n} j^{p-i} \right)=(n+1)^{p+1}-1$

Published on 25 Dec 2016 - 14:22

score 135 · Answer 4 · 2016-12-26 08:19:43

135

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Prove that:$\sum_{n=0}^{\infty}{2^{n+3}(n^2+n+\phi)\over (n+1)(2n+1)(2n+3){2n\choose n}}=\phi\pi^2+8\phi^2\pi-8\phi^3\sqrt{5}$

Published on 26 Dec 2016 - 8:19

score 274 · Answer 5 · 2026-03-28 11:35:21

274

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Show that :$\sum_{j=0}^{n}j{n\choose j}^2H_{n-j}={1\over 4}{2n\choose n}(4nH_{n}-2nH_{2n}-1)$

Published on 28 Mar 2026 - 11:35

score 1.4k · Answer 6 · 2026-03-26 04:53:33

1.4k

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Faster way to find coefficient of $x^N$ in $(1 + x + x^2 + x^3 + ... + x^A)^K$

Published on 26 Mar 2026 - 4:53

score 141 · Answer 7 · 2026-03-25 19:07:19

141

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Prove that:$\lim_{M\to \infty}\sum_{k=1}^{M}(-1)^k{M\choose k}{{\ln{(x+k)!}}\over (k+x-1)(k+x)}={\ln{x!}\over x(1-x)}$

Published on 25 Mar 2026 - 19:07

score 142 · Answer 8 · 2026-03-31 18:21:46

142

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If $\binom{n_1}{r}=\binom{n_2}{r}$,then is it true that $n_1=n_2?$

Published on 31 Mar 2026 - 18:21

score 66 · Answer 9 · 2026-04-13 06:59:49

66

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How many ways to choose K guests out of N such that M pairs are familiar?

Published on 13 Apr 2026 - 6:59

score 87 · Answer 10 · 2026-04-11 14:19:08

87

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how to evaluate this sum of binomial coefficients

Published on 11 Apr 2026 - 14:19

score 291 · Answer 11 · 2016-12-28 12:13:20

291

Views

Help to find closed form:$\sum_{n=0}^{\infty}{2^n(n^{2k+1}+n^{2k}+n^{2k-1}+\cdots+ny+x)\over (n+1)(2n+1)(2n+3){2n\choose n}}=F(k)$

Published on 28 Dec 2016 - 12:13

score 125 · Answer 12 · 2016-12-28 14:54:40

125

Views

Proving inequality $\frac{4^m}{4\sqrt{m}}\le\binom{2m}{m}$ only with Chebyshev inequality

Published on 28 Dec 2016 - 14:54

score 114 · Answer 13 · 2016-12-28 22:28:24

114

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Show that:$\int_{0}^{1}(1-\sqrt[k]{x})^ndx={1\over {k+n\choose n}}$

Published on 28 Dec 2016 - 22:28

score 76 · Answer 14 · 2026-03-31 14:32:55

76

Views

Proving the inequality

Published on 31 Mar 2026 - 14:32

score 98 · Answer 15 · 2026-03-28 11:35:20

98

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Show that : $\sum_{k=0}^{n}{(-1)^k{n\choose k}\over (k+1)^2}={1+\sqrt{1+4(n+1)^2H_nH_{n+1}}\over 2(n+1)^2}$, for $n\ge0$

Published on 28 Mar 2026 - 11:35

List Question

Is there a deep connection between the urn problem and the sum of powers?

To prove the inequality:- $\frac{4^m}{2\sqrt{m}}\le\binom{2m}{m}\le\frac{4^m}{\sqrt{2m+1}}$

Prove that $\sum _{i=0}^{p} \left( \binom {p+1}{i+1} \times \sum _{j=1}^{n} j^{p-i} \right)=(n+1)^{p+1}-1$

Prove that:$\sum_{n=0}^{\infty}{2^{n+3}(n^2+n+\phi)\over (n+1)(2n+1)(2n+3){2n\choose n}}=\phi\pi^2+8\phi^2\pi-8\phi^3\sqrt{5}$

Show that :$\sum_{j=0}^{n}j{n\choose j}^2H_{n-j}={1\over 4}{2n\choose n}(4nH_{n}-2nH_{2n}-1)$

Faster way to find coefficient of $x^N$ in $(1 + x + x^2 + x^3 + ... + x^A)^K$

Prove that:$\lim_{M\to \infty}\sum_{k=1}^{M}(-1)^k{M\choose k}{{\ln{(x+k)!}}\over (k+x-1)(k+x)}={\ln{x!}\over x(1-x)}$

If $\binom{n_1}{r}=\binom{n_2}{r}$,then is it true that $n_1=n_2?$

How many ways to choose K guests out of N such that M pairs are familiar?

how to evaluate this sum of binomial coefficients

Help to find closed form:$\sum_{n=0}^{\infty}{2^n(n^{2k+1}+n^{2k}+n^{2k-1}+\cdots+ny+x)\over (n+1)(2n+1)(2n+3){2n\choose n}}=F(k)$

Proving inequality $\frac{4^m}{4\sqrt{m}}\le\binom{2m}{m}$ only with Chebyshev inequality

Show that:$\int_{0}^{1}(1-\sqrt[k]{x})^ndx={1\over {k+n\choose n}}$

Proving the inequality

Show that : $\sum_{k=0}^{n}{(-1)^k{n\choose k}\over (k+1)^2}={1+\sqrt{1+4(n+1)^2H_nH_{n+1}}\over 2(n+1)^2}$, for $n\ge0$

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