Arithmetic - Geometric Series Duality

Algebra Level 3

n=1n3n=13+29+327+481+\large \sum _{ n=1 }^{ \infty }{ \frac { n }{ { 3 }^{ n } } } =\frac { 1 }{ 3 } +\frac { 2 }{ 9 } +\frac { 3 }{ 27 } +\frac { 4 }{ 81 } +\cdots

If the sum of the series above is in the form of ab\dfrac ab, where aa and bb are coprime positive integers, compute a×ba\times b.

If you think the series diverges, enter the answer as -1.

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