Find the sum
\[0.1 + 0.02 + 0.003 + 0.0004 + \dots + \frac{n}{10^n} + \dotsb.\]
Let
\[S = \frac{1}{10} + \frac{2}{10^2} + \frac{3}{10^3} + \dotsb.\]Then
\[\frac{1}{10} S = \frac{1}{10^2} + \frac{2}{10^3} + \frac{3}{10^4} + \dotsb.\]Subtracting these equations, get
\[\frac{9}{10} S = \frac{1}{10} + \frac{1}{10^2} + \frac{1}{10^3} + \dots = \frac{1/10}{1 - 1/10} = \frac{1}{9}.\]Therefore,
\[S = \boxed{\frac{10}{81}}.\]