Gauss Returns!

Algebra Level 2

$S = 0.1 + 0.02 + 0.003 + 0.0004 + \cdots + \dfrac n{10^n} + \cdots$

Given that the infinite sum $S$ can be expressed as $\frac ab$ , where $a$ and $b$ are coprime positive integers, find $a+b$ .