# ?How do you find the sum of the infinite geometric series 4 + 0.4 + 0.04 + ....?

Dec 30, 2015

$S u m = \frac{40}{9}$

#### Explanation:

${a}_{2} / {a}_{1} = \frac{0.4}{4} = \frac{4}{40} = \frac{1}{10}$
${a}_{3} / {a}_{2} = \frac{0.04}{0.4} = \frac{4}{40} = \frac{1}{10}$
$\implies r = \frac{1}{10}$ and ${a}_{1} = 4$

Sum of infinite geometric series is given by
$S u m = S = {a}_{1} / \left(1 - r\right) = \frac{4}{1 - \frac{1}{10}} = \frac{40}{10 - 1} = \frac{40}{9}$
$\implies S u m = \frac{40}{9}$