In a fluid mechanics problem, one has to deal with the following infinite sum: $$ S = \sum_{n \ge 1} \frac{2n+1}{n+1} \left( \int_0^1 P_n(x) \, \mathrm{d}x \right) \left( \int_0^1 x \left( 1-x^2\right) P_n'(x) \,\mathrm{d}x \right) \, , $$ with $P_n$ denoting the Legendre polynomial of degree $n$ and prime stands for a derivative with respect to the argument.
Numerically, it can be shown that $S \approx 0.1163$. Just for fun, i was wondering whether the exact result can be obtained analytically using standard tools that are usually applied to series and sequences.
Any help or hint is highly appreciated.
Thanks a lot!
The answer is $$ S=\frac{8}{9 \pi} - \frac{1}{6}.$$ Proof is fairly easy: get the integrals in explicit form and then sum. In principal, Gradshteyn and Ryzhik 7.113.1 can be used on every one of them. For the right-most integral, do the derivative and get it in terms of Legendre polynomials, so there will be two of them. As an intermediate step, I have $$ S=\frac{-1}{8}\sum_{n=1}^\infty \frac{2n+1}{n+1} \Big( \frac{ \Gamma(n/2)\ \sin{(n\pi/2)} }{\Gamma(3/2+n/2)} \Big)\Big(\frac{n+1}{\Gamma(1-n/2)\Gamma(5/2+n/2)} - \frac{n+1}{\Gamma(2-n/2)\Gamma(5/2+n/2)} \Big)$$ where the large parentheses indicate the integrals indicated in the question, and scaling factors of $\sqrt{\pi}$ and other constants have been combined and brought to the front. Do some simplification with Mathematica, and also let it solve for the closed form given above.