Example of a power series whose function has a finite limit at a point where the series diverges

Question

Let $\Sigma_n a_nz^n$ be a power series with convergence radius $R\in\mathbb R^*_+$ and say there exists $l$ such that $\Sigma_n a_nz^n \rightarrow l$ when $z\rightarrow R$. I believe that it is not necessary for $\Sigma_n a_nR^n$ to converge (or at least towards $l$ ?). Does anyone have an example ?

Answer 1

Take the series $f(z)=\sum_{n}z^n$. For $z$ in the disc of convergence, we have $f(z)=\frac{1}{1-z}$. Therefore, for $z\to-1$ we have $f(z)\to\frac{1}{2}$.

However, $\sum_{n}(-1)^n$ doesn't converge.

---

Heads up.

This exercise is preparing you for further results in which the answer is yes, but more requirements are asked from the power series. See, for example, Littlewood's Tauberian theorem.