1.36 Convergence of series

Let $\sum_{n=1}^{\infty}a_{n}$ be a series, and let

be the $n^{th}$ partial sum. If the sequence $(s_{n})$ of partial sums converges, so that $s_{n}\rightarrow s$ as $n\rightarrow\infty$ for some real $s$, then we say that the series converges to the sum $s$, and write

If the sequence of partial sums does not converge, then we say that the series diverges.

Sometimes we can work out $s_{n}$ explicitly and take the limit as $n\rightarrow\infty$ to find $s$. This $s$ is called the sum to infinity as it includes all the terms.