I have the definition:
Let $(a_n)$ be a sequence of real numbers. Let $s_n=a_1+a_2+...+a_n$. We say the series $a_1+a_2+...$ is convergent if the sequence of partial sums $(s_n)$ is convergent. The limit of this sequence is called the sum of series.
What does this mean in terms of applications to a series? How does one compute the limit of this sequence from the definition?
The statement that the partial sums $(s_n)$ are convergent is a statement that they form a Cauchy sequence. This means for any $\epsilon > 0$ there is an $N \in \mathbb{N}$ for which $$|s_n - s_m| < \epsilon$$ for all $n > m > N$. In particular if we rewrite $$s_n - s_m = \sum_{k=1}^n a_k - \sum_{k=1}^m a_k = a_{m+1} + a_{m+2} + \cdots + a_n = \sum_{k=m+1}^n a_k$$ Then we say a series converges if for all $\epsilon > 0$ there is an $N$ for which $$\left|\sum_{k=m+1}^n a_k\right| < \epsilon$$ for all $n>m>N$.
This definition isn't used to prove convergence, usually. That's what most of the theorems concerned with convergence are for.