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

Question

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? $\sum\limits_{k=0}^\infty \frac{\sqrt{a+n}}{n^2}$

Answer 1

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.