# How do you find the sum of the infinite geometric series 1 + 0.4 + 0.16 + 0.064 + . . .?

Nov 23, 2015

$\frac{5}{3}$

#### Explanation:

The terms you're summing are ${a}_{n} = {\left(\frac{4}{10}\right)}^{n}$. In fact,

${a}_{0} = {\left(\frac{4}{10}\right)}^{0} = 1$

${a}_{1} = {\left(\frac{4}{10}\right)}^{1} = \frac{4}{10} = 0.4$

${a}_{2} = {\left(\frac{4}{10}\right)}^{2} = \frac{16}{100} = 0.16$, and so on.

The general rule states that, if a series of the form

${\sum}_{n = 0}^{\infty} {k}^{n}$

converges, then in converges to $\setminus \frac{1}{1 - k}$

Since in your case $k = \frac{4}{10}$, the result is

$\frac{1}{1 - \frac{4}{10}} = \frac{1}{1 - \frac{2}{5}} = \frac{1}{\frac{3}{5}} = \frac{5}{3}$