Is it 0.1234...?

Algebra Level 3

101(1+0.1+0.01+0.001+)+102(2+0.2+0.02+0.002+)+103(3+0.3+0.03+0.003+)+...\begin{aligned} 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{aligned}

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

Clarification: In the nthn^\text{th} term, after the string of n1n - 1 zeroes, the number under the bar is nn.

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