# 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?

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

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

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

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