I am trying to find the radius of convergence for $f(x,t)=\sum P_n(x)t^n$ considered as a power series in $t$ as a function of $x$. Here $P_n(x)$ are the legendre polynomials on the interval $[-1,1]$. I know that each $|P_n(x)| \leq 1$. So I know that the radius of convergence is at least $1$ for all $x \in [-1,1]$ by dominating it by the geometric series. But I am wondering if the radius of convergence can increase.
More precisely I want to find the radius of convergence as a function of $x$. Any hints would be appreciated. Thanks.