Question

What is $\lim_{x\to\infty}\frac{\int_{0}^{x}\cos\{t-\cos t\}dt}{x}$?

score 352 · Answer 1 · 2026-03-30 02:07:15

352

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What is $\lim_{x\to\infty}\frac{\int_{0}^{x}\cos\{t-\cos t\}dt}{x}$?

Published on 30 Mar 2026 - 2:07

score 112 · Answer 2 · 2023-11-18 21:25:26

112

Views

Closed form of $\prod_{n=0}^{\infty}\frac1{1+x^{2^n}}$

Published on 18 Nov 2023 - 21:25

score 46 · Answer 3 · 2026-03-30 13:25:08

46

Views

$f_c(x) = \sum_{k = 0}^{\infty} a_k x^k =a_0 + a_1x + a_2x^2 + ...$ has no closed form apart from rational functions?

Published on 30 Mar 2026 - 13:25

score 93 · Answer 4 · 2023-11-20 04:35:57

93

Views

Compute ${\sum_{n=1}^{+\infty}(-1)^n\left(\sum_{k=1}^{n}\frac{1}{2k-1}-\frac{ \ln n}{2}-\frac{\gamma}{2}-\ln 2\right)}$

Published on 20 Nov 2023 - 4:35

score 88 · Answer 5 · 2023-11-20 12:07:11

88

Views

Find the interval of convergence $\sum_{n\geq1}\left(\ln\frac{1}{2}+1-\frac{1}{2}+\frac{1}{3}-\cdots-\frac{(-1)^n}{n}\right)x^n$

Published on 20 Nov 2023 - 12:07

score 113 · Answer 6 · 2023-11-24 08:26:52

113

Views

Evaluate $\prod_{n\geq2}\left(\frac{4}{e^2}\left(1+\frac{1}{n}\right)^{2n+1}\frac{n^2-1}{4n^2-1}\right)$

Published on 24 Nov 2023 - 8:26

score 182 · Answer 7 · 2023-11-24 09:48:30

182

Views

Calculate $\sum\limits_{n = - \infty }^\infty {\frac{{\log \left( {{{\left( {n + \frac{1}{3}} \right)}^2}} \right)}}{{n + \frac{1}{3}}}} $

Published on 24 Nov 2023 - 9:48

score 79 · Answer 8 · 2023-11-26 09:44:02

79

Views

Limit : $\lim_{n\to+\infty}a^n(n-\zeta(2)-\zeta(3)-\cdots-\zeta(n))$

Published on 26 Nov 2023 - 9:44

score 176 · Answer 9 · 2023-11-26 14:15:55

176

Views

Show that $\sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{k^2} \sum_{n=1}^k \frac{1}{n}=\frac{5\zeta(3)}{8}$

Published on 26 Nov 2023 - 14:15

score 186 · Answer 10 · 2023-11-27 09:08:31

186

Views

Show that $\sum_ {k=1}^{\infty}\dfrac{\zeta(2k)-\zeta(3k)}{k}=\ln\left(\frac{2\cosh\left(\sqrt{3}\pi/2\right)}{3\pi}\right)$

Published on 27 Nov 2023 - 9:08

score 47 · Answer 11 · 2026-04-01 05:22:11

47

Views

Make a new sequence (bn) by swapping values a0 and a1 of an old sequence (an) then figure out the closed sequence for bn

Published on 01 Apr 2026 - 5:22

score 94 · Answer 12 · 2026-03-28 02:23:49

94

Views

About $f(x) = \frac{\sum_{n=1}^{\infty} \sin^2(x/n)}{x}$

Published on 28 Mar 2026 - 2:23

score 139 · Answer 13 · 2023-11-29 20:07:18

139

Views

Show that $\sum_{n=1}^{\infty} \frac{(-1)^n (\psi(n) - \psi(2n))}{n} = \frac{\pi^2}{16} + \left(\frac{\ln(2)}{2}\right)^2$

Published on 29 Nov 2023 - 20:07

score 206 · Answer 14 · 2023-11-30 17:15:43

206

Views

Calculate $\int_0^\infty \frac{\sin\left(x^2 - \arctan\left(\frac{1}{x^2}\right)\right)}{\sqrt{1 + x^4}} \, dx $

Published on 30 Nov 2023 - 17:15

score 102 · Answer 15 · 2026-03-25 06:06:57

102

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Evaluate $\sum_{n=1}^{\infty} (-1)^{n+1} H_n \left( \frac{1}{n+1} - \frac{1}{n+3} + \frac{1}{n+5} - \ldots \right)$

Published on 25 Mar 2026 - 6:06

List Question

What is $\lim_{x\to\infty}\frac{\int_{0}^{x}\cos\{t-\cos t\}dt}{x}$?

Closed form of $\prod_{n=0}^{\infty}\frac1{1+x^{2^n}}$

$f_c(x) = \sum_{k = 0}^{\infty} a_k x^k =a_0 + a_1x + a_2x^2 + ...$ has no closed form apart from rational functions?

Compute ${\sum_{n=1}^{+\infty}(-1)^n\left(\sum_{k=1}^{n}\frac{1}{2k-1}-\frac{ \ln n}{2}-\frac{\gamma}{2}-\ln 2\right)}$

Find the interval of convergence $\sum_{n\geq1}\left(\ln\frac{1}{2}+1-\frac{1}{2}+\frac{1}{3}-\cdots-\frac{(-1)^n}{n}\right)x^n$

Evaluate $\prod_{n\geq2}\left(\frac{4}{e^2}\left(1+\frac{1}{n}\right)^{2n+1}\frac{n^2-1}{4n^2-1}\right)$

Calculate $\sum\limits_{n = - \infty }^\infty {\frac{{\log \left( {{{\left( {n + \frac{1}{3}} \right)}^2}} \right)}}{{n + \frac{1}{3}}}} $

Limit : $\lim_{n\to+\infty}a^n(n-\zeta(2)-\zeta(3)-\cdots-\zeta(n))$

Show that $\sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{k^2} \sum_{n=1}^k \frac{1}{n}=\frac{5\zeta(3)}{8}$

Show that $\sum_ {k=1}^{\infty}\dfrac{\zeta(2k)-\zeta(3k)}{k}=\ln\left(\frac{2\cosh\left(\sqrt{3}\pi/2\right)}{3\pi}\right)$

Make a new sequence (bn) by swapping values a0 and a1 of an old sequence (an) then figure out the closed sequence for bn

About $f(x) = \frac{\sum_{n=1}^{\infty} \sin^2(x/n)}{x}$

Show that $\sum_{n=1}^{\infty} \frac{(-1)^n (\psi(n) - \psi(2n))}{n} = \frac{\pi^2}{16} + \left(\frac{\ln(2)}{2}\right)^2$

Calculate $\int_0^\infty \frac{\sin\left(x^2 - \arctan\left(\frac{1}{x^2}\right)\right)}{\sqrt{1 + x^4}} \, dx $

Evaluate $\sum_{n=1}^{\infty} (-1)^{n+1} H_n \left( \frac{1}{n+1} - \frac{1}{n+3} + \frac{1}{n+5} - \ldots \right)$

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