# What is a divergent infinite series?

Sep 18, 2014

The sum $S$ of a series ${\sum}_{k = 1}^{\infty} {a}_{k}$ is defined as
$S = {\lim}_{n \to \infty} {S}_{n}$,
where ${S}_{n}$ is the nth partial sum defined as
${S}_{n} = {a}_{1} + {a}_{2} + {a}_{3} + \cdots + {a}_{n} = {\sum}_{k = 1}^{n} {a}_{k}$.

So, a series is called divergent when the limit (the sum)

$S = {\lim}_{n \to \infty} {S}_{n}$

does not exist (including infinite limits).