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

$$\lim_{x \to r^{-}} \sum a_{n} x^{n} = \sum a_{n} r^{n} = \sum (\lim_{x \to r^{-}} a_{n} x^{n})$$