Mysterious Sum

If we have the expression

1+2x+3x2+4x3+5x4+ 1+ 2x + 3x^2 + 4x^3 + 5x^4 + \cdots The value of 1+0.2+0.03+0.004+1+0.2+0.03+0.004+\cdots can be represented as ab\dfrac ab, where aa and bb are coprime positive integers, find a+ba+b.

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