Abel’s_Theorem

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Definition

Let $g(x) = \sum_{n=0}^{\infty} a_n x^n$ be a power series that converges at the point $x = R > 0$. Then the series converges uniformly on $[0, R]$, and $g$ is continuous on $[0, R]$.

In particular, $\lim_{x \to R^-} g(x) = g(R) = \sum_{n=0}^{\infty} a_n R^n$.