Frobenius & Special Functions · Superchallenge problem

Using the Legendre generating function $\dfrac{1}{\sqrt{1 - 2xt + t^2}} = \displaystyle\sum_{n\ge0}P_n(x)\,t^n$, compute $\displaystyle\sum_{n\ge0}\frac{P_n\!\left(-\tfrac12\right)}{n+1}$. (The series converges.)