Divergent series

Definition

A series $\sum a_n$ of real numbers is divergent if it is not a convergent series. Thus its sequence of partial sums has no limit in $\mathbb{R}$.

Remarks

The series $1 + 1 + 1 + \cdots$ diverges, and its partial sums tend to $+\infty$. The series $1 - 1 + 1 - 1 + \cdots$ diverges, and its partial sums oscillate between $1$ and $0$. The harmonic series diverges, although its terms tend to $0$. Summation methods, such as Cesàro summation, give values to some divergent series.

Explanations

L1 human

A divergent series has no sum in the usual sense. Its running totals grow without bound or keep jumping around.