Either Tauber's original theorem or Littlewood's improvement upon it is of this form. Let $a_n$ be a sequence of real numbers which is $o(1/n)$ (for Tauber's version) or $O(1/n)$ (for Littlewood's verion) and suppose that $\lim_{x \to 1} \sum_{n=1}^{\infty} a_n x^n = L$. Then $\sum_{n=1}^{\infty} a_n = L$.
The hard part is that $\sum_{n=1}^{\infty} a_n$ converges. Assuming the sum converges, the easier Abel's theorem tells us is must converge to $L$.