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

Since you mentioned induction:
Let $s_m = \sum_{n=2}^{m} \frac{1}{2^{n}*n}$. Then $s_m \leq 1-\frac{1}{m}$.
$P(2)$ is obvious, while $P(m) \Rightarrow P(m+1)$ reduces to
$$1-\frac{1}{m}+\frac{1}{2^{m+1}(m+1)} \leq 1- \frac{1}{m+1}$$
which is equivalent to:
$$m \leq 2^{m+1}$$
