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1×210+2×3102+3×4103+4×5104+…\large \dfrac{1 \times 2}{10} + \dfrac{2 \times 3}{10^2} + \dfrac{3 \times 4}{10^3} + \dfrac{4 \times 5}{10^4} + \ldots 101×2+1022×3+1033×4+1044×5+…
If the value of the series above can be expressed as AB\dfrac{A}{B}BA, where AAA and BBB are coprime positive integers, find the value of A+BA+BA+B.
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