# How do you find the sum of the infinite geometric series given 1-0.5+0.25-...?

Oct 23, 2016

Sum of the infinite geometric series is $\frac{2}{3}$.

#### Explanation:

As $- \frac{0.5}{1} = \frac{0.25}{-} 0.5 = - \frac{1}{2}$, the series $1 - 0.5 + 0.25 - \ldots \ldots .$ is a geometric series, whose first term $a = 1$ and common ratio $r = - \frac{1}{2}$ and $| r | < 1$.

In a geometric series where first term is $a$ and common ratio is |r}<1, sum up to ${n}^{t h}$ term ${S}_{n}$ is given by

${S}_{n} = \frac{a \left(1 - {r}^{n}\right)}{1 - r}$ and as in such infinite series as $n \to \infty$, the sum tends to $S = \frac{a}{1 - r}$

Hence, here sum of the infinite geometric series $1 - 0.5 + 0.25 - \ldots \ldots .$ is

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