Question

Integral representation of Euler-Mascheroni constant

score 271 · Answer 1 · 2026-03-24 22:29:50

271

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Integral representation of Euler-Mascheroni constant

Published on 24 Mar 2026 - 22:29

score 451 · Answer 2 · 2026-03-25 06:55:13

451

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Prove that for each $n \in \mathbb N$, $\gamma_n=1+\frac{1}{2}+\dots+\frac{1}{n}-\int_1^n \frac{1}{x}dx$ is convergent

Published on 25 Mar 2026 - 6:55

score 140 · Answer 3 · 2026-03-25 16:36:37

140

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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 - 16:36

score 555 · Answer 4 · 2026-03-25 10:00:32

555

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How do I calculate this limit $\lim _{n\to \infty }\left(e^{\sqrt{n}}\left(1-\frac{1}{\sqrt{n}}\right)^n\right)$?

Published on 25 Mar 2026 - 10:00

score 147 · Answer 5 · 2026-03-24 22:44:09

147

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Show that:$\lim\limits_{x\to \infty}{\gamma\over x}+e^{-\gamma/x}\prod\limits_{y=1}^{\infty}\left({xy\over 1+xy}\right){e^{1/(xy)}}=1$

Published on 24 Mar 2026 - 22:44

score 82 · Answer 6 · 2026-03-24 23:21:39

82

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Standard deviation of $x_1,x_2,x_3,....,x_{18}$

Published on 24 Mar 2026 - 23:21

score 441 · Answer 7 · 2026-03-24 19:02:44

441

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Alternative approaches to showing that $\gamma=\int_0^\infty \left(\frac{1}{1+x^a}-\frac{1}{e^x}\right)\,\frac1x\,dx$, $a>0$

Published on 24 Mar 2026 - 19:02

score 954 · Answer 8 · 2026-03-25 03:30:00

954

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Real Analysis Methodologies to show $\gamma =2\int_0^\infty \frac{\cos(x^2)-\cos(x)}{x}\,dx$

Published on 25 Mar 2026 - 3:30

score 478 · Answer 9 · 2026-03-25 09:08:08

478

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Mathematica gives: $\int_{0}^{\infty}{\cos(x^n)-\cos(x^{2n})\over x}\cdot{\ln{x}}\mathrm dx={12\gamma^2-\pi^2\over 2(4n)^2}$

Published on 25 Mar 2026 - 9:08

score 559 · Answer 10 · 2026-03-25 10:04:19

559

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Irrationality of Euler's constant $\gamma=0.577216...$

Published on 25 Mar 2026 - 10:04

score 84 · Answer 11 · 2026-03-24 18:08:14

84

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Constants connection of $\gamma$ and $\phi$

Published on 24 Mar 2026 - 18:08

score 208 · Answer 12 · 2026-03-25 00:17:03

208

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How can we show that :- $\sum_{n=1}^{\infty}\left({\phi^2+\gamma\over n+1}-{\phi+\gamma\over n}-\ln{n+1\over n}\right)=-\phi^2$

Published on 25 Mar 2026 - 0:17

score 91 · Answer 13 · 2026-03-25 01:25:44

91

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how to show $-\int_{-\infty }^{\infty }\!b{{\rm e}^{b-{{\rm e}^{b}}}}\,{\rm d}b= \gamma$?

Published on 25 Mar 2026 - 1:25

score 640 · Answer 14 · 2026-03-25 10:35:25

640

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Relationship between Euler's number, Uniform distribution and geometric mean

Published on 25 Mar 2026 - 10:35

score 197 · Answer 15 · 2026-03-24 23:40:03

197

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Euler Identity Derivation-Is this allowed?

Published on 24 Mar 2026 - 23:40

List Question

Integral representation of Euler-Mascheroni constant

Prove that for each $n \in \mathbb N$, $\gamma_n=1+\frac{1}{2}+\dots+\frac{1}{n}-\int_1^n \frac{1}{x}dx$ is convergent

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

How do I calculate this limit $\lim _{n\to \infty }\left(e^{\sqrt{n}}\left(1-\frac{1}{\sqrt{n}}\right)^n\right)$?

Show that:$\lim\limits_{x\to \infty}{\gamma\over x}+e^{-\gamma/x}\prod\limits_{y=1}^{\infty}\left({xy\over 1+xy}\right){e^{1/(xy)}}=1$

Standard deviation of $x_1,x_2,x_3,....,x_{18}$

Alternative approaches to showing that $\gamma=\int_0^\infty \left(\frac{1}{1+x^a}-\frac{1}{e^x}\right)\,\frac1x\,dx$, $a>0$

Real Analysis Methodologies to show $\gamma =2\int_0^\infty \frac{\cos(x^2)-\cos(x)}{x}\,dx$

Mathematica gives: $\int_{0}^{\infty}{\cos(x^n)-\cos(x^{2n})\over x}\cdot{\ln{x}}\mathrm dx={12\gamma^2-\pi^2\over 2(4n)^2}$

Irrationality of Euler's constant $\gamma=0.577216...$

Constants connection of $\gamma$ and $\phi$

How can we show that :- $\sum_{n=1}^{\infty}\left({\phi^2+\gamma\over n+1}-{\phi+\gamma\over n}-\ln{n+1\over n}\right)=-\phi^2$

how to show $-\int_{-\infty }^{\infty }\!b{{\rm e}^{b-{{\rm e}^{b}}}}\,{\rm d}b= \gamma$?

Relationship between Euler's number, Uniform distribution and geometric mean

Euler Identity Derivation-Is this allowed?

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