Question

How can one show that $\int_{0}^{\infty}\left({1\over 1+nx^n}-e^{-nx^n}\right)\cdot{\mathrm dx\over x^{1+n}}=1-\gamma?$

score 318 · Answer 1 · 2026-03-21 17:02:35

318

Views

How can one show that $\int_{0}^{\infty}\left({1\over 1+nx^n}-e^{-nx^n}\right)\cdot{\mathrm dx\over x^{1+n}}=1-\gamma?$

Published on 21 Mar 2026 - 17:02

score 93 · Answer 2 · 2026-03-21 18:32:05

93

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How does one show that $\sum_{k=2}^{\infty}{(-1)^k\over k+1}\cdot{ \lceil \log_2(k) \rceil}=1-2\gamma?$

Published on 21 Mar 2026 - 18:32

score 272 · Answer 3 · 2026-03-22 04:21:30

272

Views

How does one verify that $\int_{0}^{1}x\ln[{\ln{x}}\ln(1-x)]\mathrm dx=-\gamma?$

Published on 22 Mar 2026 - 4:21

score 341 · Answer 4 · 2026-03-22 05:03:42

341

Views

Computing the integral $\int \limits_{1}^{\infty}\left(\frac{1}{\lfloor{x}\rfloor}-\frac{1}{x}\right)$ ....

Published on 22 Mar 2026 - 5:03

score 256 · Answer 5 · 2026-03-22 01:17:12

256

Views

How can we show that $\int_{0}^{1}(\ln{x})^n\ln(-\ln{x})\mathrm dx=(-1)^n(a-b\gamma)\cdot{n!\over b}?$

Published on 22 Mar 2026 - 1:17

score 321 · Answer 6 · 2026-03-22 11:25:38

321

Views

How can one verify the closed form for $\int_{0}^{\infty}{\ln{x}\over \cosh^{2n}{x}}\mathrm dx?$

Published on 22 Mar 2026 - 11:25

score 175 · Answer 7 · 2026-03-22 13:56:58

175

Views

How does one show that $\int_{0}^{\pi/4}\sin(2x)\ln{(\ln^2{\cot{x}})}\mathrm dx=\ln{\pi\over4}-\gamma?$

Published on 22 Mar 2026 - 13:56

score 1k · Answer 8 · 2026-03-21 23:09:16

1k

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Interesting formula for Euler-Mascheroni constant?

Published on 21 Mar 2026 - 23:09

score 162 · Answer 9 · 2026-03-22 05:26:00

162

Views

Integral representation of Euler's Gamma constant

Published on 22 Mar 2026 - 5:26

score 49 · Answer 10 · 2026-03-25 04:40:50

49

Views

Find the value of the sum with accuracy up to $o(1)$.

Published on 25 Mar 2026 - 4:40

score 117 · Answer 11 · 2026-03-22 09:55:05

117

Views

How does one show that $S_1=S_2$ and has the closed form of $1?$

Published on 22 Mar 2026 - 9:55

score 110 · Answer 12 · 2026-03-22 10:08:51

110

Views

Integral of FractionalPart function

Published on 22 Mar 2026 - 10:08

score 81 · Answer 13 · 2026-03-22 09:20:35

81

Views

What happens if I divide $x$ by $\infty$?

Published on 22 Mar 2026 - 9:20

score 130 · Answer 14 · 2026-03-22 00:03:28

130

Views

How can one demonstrate that $\lim_{n\to \infty} \sum_{k=1}^{n}{2\over 2k-1}-\ln{4n}=\gamma$ is correct?

Published on 22 Mar 2026 - 0:03

score 54 · Answer 15 · 2026-03-25 09:23:34

54

Views

Can someone help me derive this equation using Euler's formula?

Published on 25 Mar 2026 - 9:23

List Question

How can one show that $\int_{0}^{\infty}\left({1\over 1+nx^n}-e^{-nx^n}\right)\cdot{\mathrm dx\over x^{1+n}}=1-\gamma?$

How does one show that $\sum_{k=2}^{\infty}{(-1)^k\over k+1}\cdot{ \lceil \log_2(k) \rceil}=1-2\gamma?$

How does one verify that $\int_{0}^{1}x\ln[{\ln{x}}\ln(1-x)]\mathrm dx=-\gamma?$

Computing the integral $\int \limits_{1}^{\infty}\left(\frac{1}{\lfloor{x}\rfloor}-\frac{1}{x}\right)$ ....

How can we show that $\int_{0}^{1}(\ln{x})^n\ln(-\ln{x})\mathrm dx=(-1)^n(a-b\gamma)\cdot{n!\over b}?$

How can one verify the closed form for $\int_{0}^{\infty}{\ln{x}\over \cosh^{2n}{x}}\mathrm dx?$

How does one show that $\int_{0}^{\pi/4}\sin(2x)\ln{(\ln^2{\cot{x}})}\mathrm dx=\ln{\pi\over4}-\gamma?$

Interesting formula for Euler-Mascheroni constant?

Integral representation of Euler's Gamma constant

Find the value of the sum with accuracy up to $o(1)$.

How does one show that $S_1=S_2$ and has the closed form of $1?$

Integral of FractionalPart function

What happens if I divide $x$ by $\infty$?

How can one demonstrate that $\lim_{n\to \infty} \sum_{k=1}^{n}{2\over 2k-1}-\ln{4n}=\gamma$ is correct?

Can someone help me derive this equation using Euler's formula?

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