Consider the generating function of Legendre polynomials: $$\frac{1}{\sqrt{1 - 2xt + t^2}} = \sum\limits^{\infty}_{n=0} P_n(x)t^n$$ Is it true that for $0<x<1, t=1$ series of Legendre polynomials converges to the function on the left-hand side, i.e.
$$\frac{1}{\sqrt{2-2x}} \stackrel{?}{=} \sum\limits^{\infty}_{n=0} P_n(x)$$
I know that the radius of convergence of $\sum\limits^{\infty}_{n=0} P_n(x)t^n$ equals to one so we need to figure out behaviour on the boundary of the disk of convergence.