# How do you find the sum of the infinite geometric series 4 - 2 + 1 – 1/2 + . . .?

Jan 21, 2016

$\frac{8}{3}$

#### Explanation:

${a}_{2} / {a}_{1} = \frac{- 2}{4} = - \frac{1}{2}$

${a}_{3} / {a}_{2} = \frac{1}{-} 2 = - \frac{1}{12}$

$\implies$ common ratio$= r = - \frac{1}{2}$ and first term$= {a}_{1} = 4$

Sum of infinite geometric series is given by

$S u m = {a}_{1} / \left(1 - r\right)$

$\implies S u m = \frac{4}{1 - \left(- \frac{1}{2}\right)} = \frac{4}{1 + \frac{1}{2}} = \frac{8}{2} + 1 = \frac{8}{3}$

$\implies S = \frac{8}{3}$

Hence the sum of the given given geometric series is $\frac{8}{3}$.