Convergence/Divergence of Complex Series $\sum\limits_{n=1}^{\infty} \frac{n(2+i)^n}{2^n}$

Your approach is good. You can alternatively solve it through the root test. One has that \begin{align*} |a_{n}| = \left|\frac{n(2+i)^{n}}{2^{n}}\right| = n\left(\frac{\sqrt{5}}{2}\right)^{n} \Longrightarrow \limsup_{n\to\infty}|a_{n}|^{1/n} = \limsup_{n\to\infty}n^{1/n}\left(\frac{\sqrt{5}}{2}\right) = \frac{\sqrt{5}}{2} > 1 \end{align*}

Thus the given series diverges.

Hopefully this helps.