# How do you find the sum of the infinite geometric series 2 + 1.5 + 1.125 + 0.8437 +…?

Jan 20, 2016

$2 + 1.5 + 1.125 + 0.84375 + \ldots = 8$

#### Explanation:

The sum of an infinite geometric series with initial value ${a}_{1}$ and ratio $r , \left\mid r \right\mid < 1$ is given by the formula:
$\textcolor{w h i t e}{\text{XXX}} \Sigma {a}_{i} = {a}_{1} / \left(1 - r\right)$

Since for the given series:
$\textcolor{w h i t e}{\text{XXX}} \frac{1.5}{2} = \frac{1.125}{1.5} = \frac{0.84375}{1.125} = 0.75$
$r = 0.75$

and
$\textcolor{w h i t e}{\text{XXX}} \Sigma {a}_{i} = \frac{2}{1 - 0.75} = 8$

Note: since we were told this was a geometric series I felt justified in replacing $0.8437$ with $0.84375$