Show that $\sum_{j=1}^{\infty}\frac{\sqrt{j+1}-\sqrt{j}}{j+1}$ is convergent or divergent.

Question

How would I show that the following series converges or diverges.

$\sum_{j=1}^{\infty}\frac{\sqrt{j+1}-\sqrt{j}}{j+1}$

I am not sure what is the best test to show that it converges. Apparently it converges.

Answer 1

You may write $$ \sum_{j=1}^{\infty}\frac{\sqrt{j+1}-\sqrt{j}}{j+1}=\sum_{j=1}^{\infty}\frac1{(j+1){(\sqrt{j+1}+\sqrt{j}})} $$ and observe that, as $j \to \infty$, $$ \frac1{(j+1){(\sqrt{j+1}+\sqrt{j}})} \sim \frac1{2j^{3/2}} $$ leading to the convergence of the initial series.

Answer 2

$$ \frac{1}{\sqrt{j}}-\frac{1}{\sqrt{j+1}} = \frac{\sqrt{j+1}-\sqrt{j}}{\sqrt{j^2+j}}\geq\frac{\sqrt{j+1}-\sqrt{j}}{j+1}\geq 0 \tag{1}$$ hence: $$ 0 \leq \sum_{j\geq 1}\frac{\sqrt{j+1}-\sqrt{j}}{j+1}\leq\sum_{j\geq 1}\left(\frac{1}{\sqrt{j}}-\frac{1}{\sqrt{j+1}}\right)=\color{red}{1}.\tag{2}$$