Answer
Divergent
Work Step by Step
Since, we have $\Sigma_{n=1}^\infty \dfrac{2^n-n}{n 2^n}$
and $\Sigma_{n=1}^\infty \dfrac{2^n-n}{n 2^n}= \Sigma_{n=1}^\infty\dfrac{2^{n}}{n2^n} -\Sigma_{n=1}^\infty\dfrac{n}{n2^n}$
or, $=\Sigma_{n=1}^\infty\dfrac{1}{n} -\Sigma_{n=1}^\infty (\dfrac{1}{2})^n$
Thus, the series $\Sigma_{n=1}^\infty\dfrac{1}{n}$ is a divergent p-series with $p=1$ and $\Sigma_{n=1}^\infty (\dfrac{1}{2})^n$ is a convergent series because it is a geometric convergent series with common ratio $r =\dfrac{1}{2} \lt 1$
Hence, the sum of the two series diverges.
