Abel’s limit theorem

Suppose that $\sum a_n{x}^n$ has a radius of convergence $r$ and that $\sum a_n{r}^n$ is convergent. Then

$$\underset{x\to r^{-}}{lim}\sum a_n{x}^n=\sum a_n{r}^n=\sum (\underset{x\to r^{-}}{lim}{a}_n{x}^n)$$