Abel's Theorem

We have just looked at Differentiation and Integration of Power Series, and now we will look at an extremely important theorem known as Abel's theorem.

Theorem: The function of a power series $f(x) = \sum_{n=0}^{\infty} a_nx^n$ is continuous on the entire interval of convergence for the series. If for some radius of convergence $R > 0$, if $\sum_{n=0}^{\infty} a_n R^n$ converges, then $\lim_{x \to R-} \sum_{n=0}^{\infty} a_nx^n = \sum_{n=0}^{\infty} a_nR^n$ and if $\sum_{n=0}^{\infty} a_n (-R)^n$ converges then $\lim_{x \to -R+} \sum_{n=0}^{\infty} a_n x^n = \sum_{n=0}^{\infty} a_n(-R)^n$.

What Abel's Theorem says is that the power series representation $f(x) = \sum_{n=0}^{\infty} a_n(x - c)^n$ is continuous on the entire interval of convergence for the series, and so the limit as $x$ approaches $R$ from the left is going to equal the sum of the series $\sum_{n=0}^{\infty} a_nR^n$, and the limit as $x$ approaches $-R$ from the right is going to equal the sum of the series $\sum_{n=0}^{\infty} a_n(-R)^n$.