Abel's Theorem (power series at x=1)

Abel’s Theorem (one standard form): Let $(a_n)$ be a real or complex sequence such that the series $\sum_{n=0}^\infty a_n$ converges to $s$. For $0\le r<1$, define $f(r)=\sum_{n=0}^\infty a_n r^n,$ which converges for $|r|<1$. Then $\lim_{r\to 1^-} f(r)=s.$

This theorem connects ordinary convergence of $\sum a_n$ to the boundary behavior of the associated power series with radius $1$. It is one of the basic results explaining why power series behave well as the variable approaches the boundary from within.

Proof sketch: Let $s_n=\sum_{k=0}^n a_k$. One can rewrite (Abel summation) $\sum_{n=0}^\infty a_n r^n = (1-r)\sum_{n=0}^\infty s_n r^n.$ Since $s_n\to s$, the sequence $(s_n)$ is bounded . As $r\to 1^-$, the weights $(1-r)r^n$ form an approximate “probability distribution” concentrating on large $n$, forcing $(1-r)\sum s_n r^n \to s$.