# How do you find the sum of the infinite geometric series 2-2+2-2+..?

Dec 20, 2015

The series diverges by typical summation methods, but may converge depending on the method used.

#### Explanation:

In general, a geometric series will diverge if the common ratio $r$ between terms does not have $| r | < 1$. In this case, $r = - 1$, meaning $| r | = 1$ and so the series should diverge.

Still, let's look at this case in a little more detail.

A series ${\sum}_{k = 0}^{\infty} {a}_{k}$ may be defined as ${\lim}_{n \to \infty} {\sum}_{k = 0}^{n} {a}_{k}$.

Typically, we refer to ${\sum}_{k = 0}^{n} {a}_{k}$ as the ${n}^{t h}$ partial sum of the series (often denoted ${S}_{n}$).

For the series in question, we have

${S}_{n} = {\sum}_{k = 0}^{n} {\left(- 1\right)}^{k} \cdot 2 = \left\{\begin{matrix}2 \text{ if n is even" \\ 0" if n is odd}\end{matrix}\right.$

As the sequence $\left({S}_{n}\right)$ simply alternates between $2$ and $0$, it should be clear that ${\sum}_{k = 0}^{\infty} {\left(- 1\right)}^{k} \cdot 2 = {\lim}_{n \to \infty} {S}_{n}$ does not converge to any value
(the fact that the sequence $\left({S}_{n}\right)$ does not converge may be shown simply using an $\epsilon - \delta$ proof).

The particular series mentioned is similar to a famous series known as Grandi's Series. Like Grandi's series, it diverges in conventional summation, but may converge if another method, such as Cesàro summation, is used.

Numberphile has a nice video on the subject here