# How do you find the sum of the infinite geometric series -2+1-1/2+...?

Dec 22, 2015

$- \frac{4}{3}$

#### Explanation:

The formula for the sum of a geometric series is

$S = \frac{r}{1 - r}$

where $r$ is the ratio between the successive terms of the series.

Let's look at the terms to find out what the ratio is. Divide $1$ by $- 2$. (the second term by the first term)

The ratio is $- \frac{1}{2}$.

Indeed, divide $- \frac{1}{2}$ (3rd term) by $1$ (2nd term) and you'll also get $- \frac{1}{2}$.

So, let's plug our ratio into the formula:

$S = \frac{- \frac{1}{2}}{1 - \left(- \frac{1}{2}\right)}$
$S = - \frac{1}{3}$

Be careful here - this is the sum of a series whose first term is the ratio itself. In other words, this is the sum of the series

$- \frac{1}{2} + \frac{1}{4} - \frac{1}{8} + \frac{1}{16} - \ldots = S$

To get the correct sum, all we need to do is add $- 2 + 1$ to $S$.

$- \frac{1}{3} - 2 + 1 = - \frac{4}{3}$