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• 1,179

### Simplify in closed-form $\sum_n P_n(0)^2 r^n P_n(\cos \theta)$

To express the series as an integral, one can use the generating function for the product of two Legendre polynomials derived by Leonard C. Maximon here: \sum_{p=0}^\infty t^{p}P_p(\...
• 14.1k

• 42.1k

### Conjecture ${\large\int}_0^\infty\left[\frac1{x^4}-\frac1{2x^3}+\frac1{12\,x^2}-\frac1{\left(e^x-1\right)x^3}\right]dx=\frac{\zeta(3)}{8\pi^2}$

Utilize the known integral $$\frac1\pi\int_0^\infty \frac{\sin\frac{xt}{2\pi}}{e^t-1}dt = \frac12 \coth\frac x2-\frac1x$$ to evaluate \begin{align} &\int_0^\infty\left[\frac1{x^4}-\frac1{2x^3}+\...
• 98.1k

### How to Evaluate $\sum_{n=0}^{\infty}\frac{(-1)^n(4n+1)(2n)!^3}{2^{6n}n!^6}$

We prove in this answer Bauer's series that was known since 1859. The original proof uses Fourier-Legendre expansions as I know. This series appears in one of the letters that Ramanujan sent to Hardy. ...
• 275

### compute the following integral in closed form : $\int_0^{\frac{π}{2}}\frac{x}{(1+\sqrt 2)\sin^{2}(x)+8\cos^{2} x}dx$

Let $a=\sqrt{8(\sqrt2-1)}$\begin{align} &\int_0^{\frac{π}{2}}\frac{x}{(1+\sqrt 2)\sin^{2} (x)+8\cos^{2} x}dx\\ \overset{ibp}=& \ \frac a8\int_0^{\frac{π}{2}}\cot^{-1}\frac{\tan x}{a}dx = \...
• 98.1k

### Power series for $\sum_{n=0}^\infty(-1)^n/n!^s$ (around $s=0$)

Denoting $\,S(s)=\displaystyle \sum_{n=0}^\infty\frac{(-1)^n}{n!^s}=\sum_{n=0}^\infty(-1)^ne^{-s\ln\Gamma(n+1)}$ we can use the Lindelöf summation formula for alternating series (also, for example, ...
• 15.8k

### Show that $\int_{0}^{1} \frac{\tan^{-1}(x^2)}{\sqrt{1 - x^2}} \, dx = \frac{1}{2}\pi \tan^{-1}\left(\sqrt{\frac{1}{\sqrt{2}} - \frac{1}{2}}\right)$

Let’s start from $$I=\int_0^{\frac{\pi}{2}} \tan ^{-1}\left(\sin ^2 t\right) d t$$ with its parametrised integral $$\int_0^{\frac{\pi}{2}} \tan ^{-1}\left(a\sin ^2 t\right) d t$$ Differentiating w.r.t ...
• 20.6k

• 4,229
1 vote

### Integral in terms of Hypergeometric function

This is not an answer From a computation point of view, why not to stay with $$I=\int_0^1 \frac { B_v(2(1- s),s)} {v^{2(1- s)} \left(1+c^2 v^2\right)^s }\,dv$$ which does not seem to make too much ...
• 262k
1 vote

### How to represent $x^n$ as a sum of $P_k:= (x)(x-1)\dots(x-k+1)$?

Maybe this can be another way to solve the problem, let me know if you agree with me. You can observe that $$P_k(j)=0$$ for each $j\leq k-1$, and that $$P_k(j)=j(j-1)\cdots (j-k+1)=\binom{j}{k}k!$$ ...
• 8,603
1 vote

### Compute in closed form $\int_0^1\frac{\arctan{ax}}{\sqrt{1-x^{2}}}dx$

I would like to consider a generalization of this problem. Define the function $\mathcal{I}:\mathbb{R}\times[-1,1]\rightarrow\mathbb{R}$ via the definite integral \mathcal{I}{\left(a,b\right)}:=\...
• 30k

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