Question

Evaluate $\sum_{n=0}^{\infty} \frac{{\left(\left(n+1\right)\ln{2}\right)}^n}{2^n n!}$

score 966 · Answer 1 · 2026-04-17 08:07:21

966

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Evaluate $\sum_{n=0}^{\infty} \frac{{\left(\left(n+1\right)\ln{2}\right)}^n}{2^n n!}$

Published on 17 Apr 2026 - 8:07

score 143 · Answer 2 · 2026-04-16 15:42:43

143

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Evaluating $\lim\limits_{n \to \infty} \sqrt[n]{\frac{n!}{\sum\limits_{m=1}^n m^m}}$

Published on 16 Apr 2026 - 15:42

score 29 · Answer 3 · 2026-04-12 15:59:44

29

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Algorithms and Summations

Published on 12 Apr 2026 - 15:59

score 133 · Answer 4 · 2026-04-17 12:19:33

133

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Evaluate $\sum_{n=1}^{50}n\cdot n!$.

Published on 17 Apr 2026 - 12:19

score 47 · Answer 5 · 2026-04-13 05:15:10

47

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Can this sum of fractions of pairwise sums be computed efficiently?

Published on 13 Apr 2026 - 5:15

score 98 · Answer 6 · 2026-04-16 02:35:08

98

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Prove $(\frac{a}{b-c})^2+(\frac{b}{c-a})^2+(\frac{c}{a-b})^2 \geq 2$

Published on 16 Apr 2026 - 2:35

score 102 · Answer 7 · 2026-04-15 20:23:06

102

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Calculate $\frac{\pi}{2}-\sin(\frac{\pi}{2})+\sin(\sin(\frac{\pi}{2}))-\sin(\sin(\sin(\frac{\pi}{2})))\cdots$

Published on 15 Apr 2026 - 20:23

score 242 · Answer 8 · 2026-04-16 22:46:04

242

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Evaluate: $\sum_{n=1}^{\infty} {\left(\frac{-100}{729}\right)}^n {3n \choose n}$

Published on 16 Apr 2026 - 22:46

score 96 · Answer 9 · 2026-05-11 08:39:42

96

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integral of fractional part $\int_0^1\left\{\frac 1x\right\}dx$ convergent?

Published on 11 May 2026 - 8:39

score 132 · Answer 10 · 2026-05-11 05:23:50

132

Views

sum upto n terms where rth term is $r(r+1)2^r$

Published on 11 May 2026 - 5:23

score 144 · Answer 11 · 2026-04-10 09:45:54

144

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How to interchange the two summations and simplify?

Published on 10 Apr 2026 - 9:45

score 109 · Answer 12 · 2026-05-11 01:11:10

109

Views

Let $a, b, c>0$. Prove that $\sum \limits_{cyc}{\frac{a}{b+c}\left(\frac{b}{c+a}+\frac{c}{a+b}\right)}\le \frac{(a+b+c)^2}{2(ab+bc+ca)}$

Published on 11 May 2026 - 1:11

score 165 · Answer 13 · 2026-04-17 05:34:14

165

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Show $\sum_{k=1}^n {k+1\choose 2}{2n+1\choose n+k+1}={n\choose 1}4^{n-1}$

Published on 17 Apr 2026 - 5:34

score 73 · Answer 14 · 2026-04-12 05:48:46

73

Views

Simplify $\sum_{k=0}^{\infty} x^ky^{\frac{k(k+1)}{2}}$

Published on 12 Apr 2026 - 5:48

score 31 · Answer 15 · 2026-04-13 17:03:06

31

Views

Is the sum of partial sums $\frac{2\pi}{k}-{k \mod 2\pi}$ bounded?

Published on 13 Apr 2026 - 17:03

List Question

Evaluate $\sum_{n=0}^{\infty} \frac{{\left(\left(n+1\right)\ln{2}\right)}^n}{2^n n!}$

Evaluating $\lim\limits_{n \to \infty} \sqrt[n]{\frac{n!}{\sum\limits_{m=1}^n m^m}}$

Algorithms and Summations

Evaluate $\sum_{n=1}^{50}n\cdot n!$.

Can this sum of fractions of pairwise sums be computed efficiently?

Prove $(\frac{a}{b-c})^2+(\frac{b}{c-a})^2+(\frac{c}{a-b})^2 \geq 2$

Calculate $\frac{\pi}{2}-\sin(\frac{\pi}{2})+\sin(\sin(\frac{\pi}{2}))-\sin(\sin(\sin(\frac{\pi}{2})))\cdots$

Evaluate: $\sum_{n=1}^{\infty} {\left(\frac{-100}{729}\right)}^n {3n \choose n}$

integral of fractional part $\int_0^1\left\{\frac 1x\right\}dx$ convergent?

sum upto n terms where rth term is $r(r+1)2^r$

How to interchange the two summations and simplify?

Let $a, b, c>0$. Prove that $\sum \limits_{cyc}{\frac{a}{b+c}\left(\frac{b}{c+a}+\frac{c}{a+b}\right)}\le \frac{(a+b+c)^2}{2(ab+bc+ca)}$

Show $\sum_{k=1}^n {k+1\choose 2}{2n+1\choose n+k+1}={n\choose 1}4^{n-1}$

Simplify $\sum_{k=0}^{\infty} x^ky^{\frac{k(k+1)}{2}}$

Is the sum of partial sums $\frac{2\pi}{k}-{k \mod 2\pi}$ bounded?

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