Abel's theorem

A boundary limit theorem relating a convergent series to its associated power series near the radius 1.

Abel’s theorem

Abel’s theorem: Let $\sum_{n=0}^\infty a_n$ be a convergent series of real or complex numbers, with sum $s$ . For $0\le x<1$ , define

f(x)=\sum_{n=0}^\infty a_n x^n.

Then $f(x)$ is well-defined for every $0\le x<1$ , and

\lim_{x\to 1^-} f(x)=s.

This connects ordinary convergence of series with boundary behavior of power series and complements radius-of-convergence results such as the Cauchy–Hadamard theorem .