If $\lim_{x \to R} \sum_{n = 0}^{\infty} a_n x^n = c \in \mathbb{R}$, does $\sum_{n = 0}^{\infty} a_n R^n$ converge?

Question

Let $R \in \mathbb{R}$ be the radius of convergence of $\sum_{n = 0}^{\infty} a_n x^n.$

If $\lim_{x \to R} \sum_{n = 0}^{\infty} a_n x^n$ diverges, then $\sum_{n = 0}^{\infty} a_n R^n$ diverges by Abel's theorem.

My question is here:

If $\lim_{x \to R} \sum_{n = 0}^{\infty} a_n x^n = c \in \mathbb{R}$, does $\sum_{n = 0}^{\infty} a_n R^n$ converge?

Answer 1

Hint:

Consider $\sum_{n=0}^\infty (-1)^n x^n$ with $R=1$.