Answer
Divergent
Work Step by Step
Re-write the given series as follows: $\Sigma_{n=1}^\infty \dfrac{2^n-n}{n 2^n}= \Sigma_{n=1}^\infty\dfrac{2^{n}}{n2^n} -\Sigma_{n=1}^\infty\dfrac{n}{n2^n}$
$\implies \Sigma_{n=1}^\infty \dfrac{2^n-n}{n 2^n}=\Sigma_{n=1}^\infty\dfrac{1}{n} -\Sigma_{n=1}^\infty (\dfrac{1}{2})^n$
Here, the first part of the series, that is, $\Sigma_{n=1}^\infty\dfrac{1}{n}$ shows a divergent p-series with $p=1$ and the second part of the series, that is, $\Sigma_{n=1}^\infty (\dfrac{1}{2})^n$ shows a geometric convergent series and has common ratio $r =\dfrac{1}{2} \lt 1$.
Therefore, the sum of the series is a divergent series.
