1. **State the problem:** We want to find for which values of $p$ the series $$\sum_{n=1}^{\infty} \frac{n}{\sqrt{4+n^p}}$$ converges. 2. **Analyze the general term:** The term is $$a_n = \frac{n}{\sqrt{4+n^p}}.$$ For large $n$, the $4$ inside the square root becomes negligible compared to $n^p$, so $$a_n \sim \frac{n}{\sqrt{n^p}} = \frac{n}{n^{p/2}} = n^{1 - \frac{p}{2}}.$$ 3. **Use the p-series test:** The series behaves like $$\sum n^{1 - \frac{p}{2}}.$$ A p-series $$\sum n^{-q}$$ converges if and only if $q > 1$. 4. **Set the exponent condition:** We want $$1 - \frac{p}{2} < -1,$$ which means $$1 - \frac{p}{2} < -1 \implies -\frac{p}{2} < -2 \implies \frac{p}{2} > 2 \implies p > 4.$$ 5. **Conclusion:** The series converges if and only if $$p > 4.$$