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 \( \dfrac ab\), where \(a\) and \(b\) are coprime positive integers, find \(a+b\).

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