Arithmetic - Geometric Series Duality

Algebra Level 3

\[\large \sum _{ n=1 }^{ \infty }{ \frac { n }{ { 3 }^{ n } } } =\frac { 1 }{ 3 } +\frac { 2 }{ 9 } +\frac { 3 }{ 27 } +\frac { 4 }{ 81 } +\cdots \]

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

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

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