TITLE: Is there a formal definition of convergence of series?
QUESTION [5 upvotes]: One is often asked to check if a given series converges or not in one's first year of uni. Is there a formal definition that allows us to check this? We are only given a bunch of tests that are troublesome to remember (I've never liked cramming in math), and a Google-search only yields results regarding sequences.

REPLY [4 votes]: Let $\sum_{n=m}^\infty a_n$ be a formal infinite series. For any integer $N\geqslant m$, we define the $N^{\text{th}}$ partial sum $S_N$ of this series to be $S_N:=\sum_{n=m}^N a_n$; of course, $S_N$ is a real number. If the sequence $(S_N)_{n=m}^\infty$ converges to some limit $L$ as $N\to\infty$, then we say that the infinite series $\sum_{n=m}^\infty a_n$ is convergent, and converges to $L$; we also write $L=\sum_{n=m}^\infty a_n$, and say that $L$ is the sum of the infinite series $\sum_{n=m}^\infty a_n$. If the partial sums $S_N$ diverge, then we say that the infinite series $\sum_{n=m}^\infty a_n$ is divergent.
from Terence Tao - Analysis I.