# How do you find the sum of the infinite geometric series 3 - 2 + 4/3 - 8/9 + ...?

Nov 10, 2015

The sum of this series is $1 \frac{4}{5}$

#### Explanation:

${a}_{1} = 3$

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

Since $| q | < 1$, the sequence is convergent, so we can calculate the sum using:

$S = {a}_{1} / \left(1 - q\right) = \frac{3}{1 - \left(- \frac{2}{3}\right)} = \frac{3}{1 + \frac{2}{3}} = \frac{3}{\frac{5}{3}} = 3 \cdot \frac{3}{5} = \frac{9}{5} = 1 \frac{4}{5}$