# How do you find the sum of the infinite geometric series 0.9 + 0.09 + 0.009 +…?

Feb 13, 2016

It will be $0. \overline{9}$ which is taken as equal to 1

#### Explanation:

For instance:

$\frac{1}{3} = 0. \overline{3}$

$3 \times \frac{1}{3} = 0. \overline{9}$ but is also equal to 1, therefore both numbers are equal.

Feb 13, 2016

Apply the geometric series formula for $a = \frac{9}{10}$ and $r = \frac{1}{10}$ to find that $0.9 + 0.09 + 0.009 + \ldots = 1$

#### Explanation:

The other answer is correct, but to add how applying the geometric series formula would give the result:

For $| r | < 1$ and $a \ne 0$ we have ${\sum}_{n = 0}^{\infty} {r}^{n} = \frac{a}{1 - r}$

(for a short derivation of this formula, see this answer)

Applying this, as $| \frac{1}{10} | < 1$, we have

$0.9 + 0.09 + 0.009 + \ldots = {\sum}_{n = 0}^{\infty} \frac{9}{10} {\left(\frac{1}{10}\right)}^{n}$

$= \frac{\frac{9}{10}}{1 - \frac{1}{10}}$

$= \frac{\frac{9}{10}}{\frac{9}{10}}$

$= 1$