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?