MathItems

Given a complex-valued sequence $\{a_n\}$ we associate a sequence $\{s_n\}$, where

$$s_n=\sum_{k=1}^n a_k \; .$$

For $\{s_n\}$ we also use the symbolic expression

$$a_1 + a_2 + a_3 + \cdots$$

or, more concisely,

$$\sum_{n=1}^\infty a_n \; .$$

This last symbol we call an infinite series, or just a series. The numbers $s_n$ are called the partial sums of the series. If $\{s_n\}$ converges to $s$, we say that the series converges, and write

$$\sum_{n=1}^\infty a_n=s \; .$$

The number $s$ is called the sum of the series.

If $\{s_n\}$ diverges, the series is said to diverge.