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I want to show that the following series converges.

$$\sum_{n=1}^\infty 2^{-n/2}\sqrt{\log{(2^n)}}$$

I tried to bound it from above by a convergent sequence without success. It would be appreciated if someone could help me. Thanks four your help.

hulik

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    As an aside, it converges to $\sqrt{\log(2)}\mathrm{Li}_{\frac{-1}{2}}\left(\frac{1}{\sqrt{2}}\right)$, where $\mathrm{Li}$ is the polylogarithm.2012-03-05

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