# How do you find the sum of the convergent series 0.9-0.09+0.009-...? If the convergent series is not convergent, how do you know?

##### 1 Answer
Dec 30, 2015

Note that the common ratio has an absolute value less than one, and use the geometric series formula to find that

$0.9 - 0.09 + 0.009 - \ldots = \frac{9}{11} = 0. \overline{81}$

#### Explanation:

A geometric series is a series of the form

$a + a r + a {r}^{2} + a {r}^{3} + \ldots = {\sum}_{n = 0}^{\infty} a {r}^{n}$

where $a$ is the initial term of the series and $r$ is the common ratio between terms.

If $| r | < 1$, then the sum is given by

${\sum}_{n = 0}^{\infty} a {r}^{n} = \frac{a}{1 - r}$

An explanation of where this sum comes from is given in this answer.

For the given series, we have

$0.9 - 0.09 + 0.009 - \ldots = \frac{9}{10} + \frac{9}{10} \left(- \frac{1}{10}\right) + \frac{9}{10} {\left(- \frac{1}{10}\right)}^{2} + \ldots$

$= {\sum}_{n = 0}^{\infty} \frac{9}{10} {\left(- \frac{1}{10}\right)}^{n}$

Thus $a = \frac{9}{10}$ and $r = - \frac{1}{10}$.
As $| r | = \frac{1}{10} < 1$ we have the sum

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

$= \frac{\frac{9}{10}}{1 - \left(- \frac{1}{10}\right)}$

$= \frac{9}{11} = 0. \overline{81}$