Evaluating an Infinite Series with Roots

Evaluate: $$\frac{1/2}{1+\sqrt2}+\frac{1/4}{1+\sqrt[4]2}+\frac{1/8}{1+\sqrt[8]2}+\frac{1/16}{1+\sqrt[16]2}+\cdots$$The answer can be expressed as $\log_{\beta} \left(\dfrac{e}{\alpha}\right)$, where $\alpha$ and $\beta$ are positive integers. Find the value of $\alpha+\beta$.