Question

find the sum of the infinite geometric series.
$5 - \frac{5}{8} + \frac{5}{64} - \frac{5}{512} + \dots$

the sum of the infinite geometric series is \boxed{}.
(type an integer or a simplified fraction.)

Explanation:

Step1: Identify first term and ratio

For a geometric series \(a + ar + ar^2 + \dots\), first term \(a = 5\).
Ratio \(r=\frac{-\frac{5}{8}}{5}=-\frac{1}{8}\). Check \(|r|=\frac{1}{8}<1\), so sum exists.

Step2: Apply infinite geometric sum formula

The formula for the sum \(S\) of an infinite geometric series is \(S = \frac{a}{1 - r}\) (when \(|r| < 1\)).
Substitute \(a = 5\) and \(r = -\frac{1}{8}\):
\(S=\frac{5}{1 - (-\frac{1}{8})}=\frac{5}{1+\frac{1}{8}}=\frac{5}{\frac{9}{8}} = 5\times\frac{8}{9}=\frac{40}{9}\).

Answer:

\(\frac{40}{9}\)