user90369

Member for 7 years, 7 months
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calculus

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24

Evaluating integral $\int_0^1 \frac{x-x^2}{\sin \pi x} dx = \frac{7 \zeta(3)}{\pi^3}$

May 24, 2018

22

Product $\left(\frac31\right)^{1/2}\cdot\left(\frac75\right)^{1/6}\cdot \left(\frac{11}9\right)^{1/10}\cdot \left(\frac{15}{13}\right)^{1/14}\cdots$

Sep 26, 2017

17

Generalised Integral $I_n=\int_0^{\pi/2} \frac{x^n}{\sin ^n x} \ \mathrm{d}x, \quad n\in \mathbb{Z}^+.$

Jul 14, 2016

14

Prove $\int_0^1 \frac{4\cos^{-1}x}{\sqrt{2x-x^2}}\,dx=\frac{8}{9\sqrt{\pi}}\left(9\Gamma(3/4)^2{}_4F_3(\cdots)+\Gamma(5/4)^2{}_4F_3(\cdots)\right)$

Aug 23, 2017

12

Closed form of $\int_0^\infty \left(\frac{\arctan x}{x}\right)^ndx$

Oct 2, 2018

Questions

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8

Limit of $\sqrt{\frac{\pi}{1-x}}-\sum\limits_{k=1}^\infty\frac{x^k}{\sqrt{k}}$ when $x\to 1^-$?

May 17, 2016

8

The sum of fractional powers $\sum\limits_{k=1}^x k^t$.

Sep 2, 2016

8

convergence of $\sum\limits_{n=1}^\infty \frac{(-1)^{\lfloor \sqrt{n}\rfloor}}{\sqrt{n}}$

Apr 7, 2017

7

Equality of the sums $\sum\limits_{v=0}^k \frac{k^v}{v!}$ and $\sum\limits_{v=0}^k \frac{v^v (k-v)^{k-v}}{v!(k-v)!}$

Jul 28, 2016

5

Solutions for $\cos(\alpha)+\cos(\beta)-2\cos(\alpha+\beta)=0$ with a certain value range.

Oct 31, 2016

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calculus

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sequences-and-series

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integration

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definite-integrals

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real-analysis

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posts

Reputation

+10

Infinite product $\Gamma(\tfrac14)=\mathrm{A}^3e^{-\mathrm{G}/\pi}2^{1/6}\sqrt{\pi}\prod_{k\ge1}\left(1-\frac1{2k}\right)^{(-1)^k k}$

+10

Evaluating integral $\int_0^1 \frac{x-x^2}{\sin \pi x} dx = \frac{7 \zeta(3)}{\pi^3}$

+10

Evaluating $\lim\limits_{n\to\infty} e^{-n} \sum\limits_{k=0}^{n} \frac{n^k}{k!}$

+10

Prove inequality $\arccos \left( \frac{\sin 1-\sin x}{1-x} \right) \leq \sqrt{\frac{1+x+x^2}{3}}$

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