Can we have an example of two distinct power series $\sum_{n=0}^\infty a_nx^n$ and $\sum_{n=0}^\infty b_n x^n$ with the radius of convergence equal $1$ and there exists a sequence $\{x_i\}_{i\in\mathbb{N}}$ in $(-1,1)$ so that $$\sum_{n=0}^\infty a_nx_i^n = \sum_{n=0}^\infty b_nx_i^n$$ for $i\in \mathbb{N}$?
Theorem 8.5 in Rudin's Principles of Mathematical Analysis says we don't have such an example if $\{x_i\}_{i\in\mathbb{N}}$ has a limit point in $(-1,1)$. What if $x_i\nearrow 1$?
Consider the power series of the following two functions around the origin $$ f(z)= (1-z)^{-1} \cos( (1-z)^{-1}), g(z)=42 f(z).$$ The only singularity of $f,g$ is at $z=1$, hence the associated power series around the origin has radius of convergence equal to $1$. Furthermore, the functions are different, but have the same zeros, which accumulate at $z=1$, thus we get a counterexample.