Is It A Converging Series?

Algebra Level 4

\[S=0.1+0.04+0.009+0.0016+\cdots\] \(S\) represents an arithmetic-geometric progression.

If \(S\) can be represented in the form \(\dfrac{a}{b}\), where \(a\) and \(b\) are coprime positive integers, find \(a+b\)

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