Complex generating function for Legendre polynomials

Question

I want to clarify, does the series $$ \sum_{n=0}^{\infty} P_n(x) e^{int} \qquad (1) $$ converge? I know that for all complex $z$, such that $|z|<1$, we have formula $$ \sum_{n=0}^{\infty} P_n(x) z^n = g(x,z) = \frac{1}{\sqrt{1-2xz+z^2}} . \qquad (2) $$ But function $g(x,z)$ has singularities on circle $|z|=1$ and it seems that expression $(2)$ is not applicable in my case $(1)$.