Sequence and Series 1

Algebra Level 4

\[\dfrac{1}{2}\left( \dfrac{1}{2}+\frac{1}{3} \right) + \dfrac{2}{2}\left( \dfrac{1}{2^2}+\frac{1}{3^2} \right) + \dfrac{3}{2}\left( \dfrac{1}{2^3}+\frac{1}{3^3} \right) + \cdots = \displaystyle \sum^{\infty}_{n=1}\dfrac{n}{2}\left( \dfrac{1}{2^n} + \dfrac{1}{3^n}\right) \]

If the value of the series above is equal to \(\dfrac AB\), where \(A\) and \(B\) are positive integers, find \(A+B\).

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