Is it 0.1234...?

Algebra Level 3

\[\begin{align*} 10^{-1}(1 + 0.1 + 0.01& + 0.001 + \cdots) + \\ 10^{-2}(2 + 0.2 + 0.02& + 0.002 + \cdots) + \\ 10^{-3}(3 + 0.3 + 0.03& + 0.003 + \cdots) + \\ &. \\ &. \\ &. \\ \end{align*}\]

The above sum can be represented as \(\left(\dfrac{a}{b}\right)^{2}\), where \(a\) and \(b\) are coprime positive integers. Find the value of \(a+b\).

Clarification: In the \(n^\text{th}\) term, after the string of \(n - 1\) zeroes, the number under the bar is \(n\).

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