An Alternating Weighted Series

Determine the value of $a+b$ if: $$\frac{a}{b} =\left ( \frac{1}{2}\times 1 \right )- \left (\frac{3}{10}\times 2 \right ) + \left (\frac{9}{50}\times 3 \right ) - \left (\frac{27}{250}\times 4 \right ) + \dots $$where $a$ and $b$ are coprime positive integers.