Abel's theorem

In mathematics, Abel's theorem for power series relates a limit of a power series to the sum of its coefficients. It is named after Norwegian mathematician Niels Henrik Abel.

Let

be a power series with real coefficients a_k with radius of convergence 1. Suppose that the series ${\displaystyle \sum _{k=0}^{\infty }a_{k}\!}$ converges. Then G(x) is continuous from the left at ${\displaystyle x=1}$, i.e.

The same theorem holds for complex power series ${\displaystyle G(z)=\sum _{k=0}^{\infty }a_{k}z^{k}}$, provided that ${\displaystyle z\rightarrow 1}$ within a Stolz sector, that is, a region of the open unit disk where

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