Gauss Returns!

Algebra Level 2

S=0.1+0.02+0.003+0.0004++n10n+ S = 0.1 + 0.02 + 0.003 + 0.0004 + \cdots + \dfrac n{10^n} + \cdots

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

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