# How do you find the sum of the convergent series 0.1+0.01+0.001+...? If the convergent series is not convergent, how do you know?

Mar 22, 2016

$x = \frac{1}{9}$

#### Explanation:

Take

$x = 0.1 + 0.01 + .001 + 0.0001 \ldots .$

But

$0.1 + \frac{x}{10} = 0.1 + 0.01 + .001 + 0.0001 \ldots .$

Therefore:

$x = 0.1 + \frac{x}{10}$

$10 x = 1 + x$

$9 x = 1$

$x = \frac{1}{9}$