# Pointwise convergence of $\sum\limits_{k=0}^\infty \frac{\sqrt{a+n}}{n^2}$

I need to prove that this series is pointwise convergent for $$a>0$$, but the ratio test, root test and the convergent minorant $$\frac{1}{n^2}$$ are inconclusive, so how would I be able to prove this? $$\sum\limits_{k=0}^\infty \frac{\sqrt{a+n}}{n^2}$$

The series $$\sum_{k=0}^{\infty} \frac{\sqrt{a+n}}{n^2}$$ is made of positive terms, therefore we can look at the asymptotic behaviour of the general term to decide whether it converges or not. We see that $$\lim_{n \to +\infty} \frac{\sqrt{a+n}}{n^2} \cdot n^{\frac{3}{2}} = \lim_{n \to +\infty} \sqrt{\frac{a+n}{n}} = 1$$ i.e. the general term is asymptotic to $$\frac{1}{n^{3/2}}$$. As the series $$\sum_{k=0}^{\infty} \frac{1}{n^{3/2}}$$ converges, we deduce that the initial series converges as well.