Want to show $\sum_{n=2}^{\infty} \frac{1}{2^{n}*n}$ converges Want to show $\sum_{n=2}^{\infty} \frac{1}{2^{n}*n}$ converges. I am trying to show this by showing that the partial sums are bounded. I have tried doing this by induction but am not seeing how to pass the inductive assumption part. Do I need to instead look for a closed form? thanks

Go back to the Cauchy test for convergence of a sequence.
Let $S_n = \sum_{k=2}^{n} \frac{1}{k2^k}$.
If $m < n$,
$S_n-S_m = \sum_{k=m+1}^{n} \frac{1}{k2^k}
< \sum_{k=m+1}^{n} \frac{1}{2^k}
< \frac{1}{2^m}
$
and this $\to 0$
as $m \to \infty$.
Therefore the sequence converges.
