# 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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