How can I show if this series diverges or converges: $\sum_{n=1}^\infty \frac{1}{\sqrt{n+2}+\sqrt{n}}$?

Question

So, I am trying to show if $\sum_{n=1}^\infty \frac{1}{\sqrt{n+2}+\sqrt{n}}$ diverges or converges through the comparison test. However, I am having difficulty finding something to compare it to. I know that it diverges, but I am unsure of a smaller series that also diverges. Thanks in advance.

Answer 1

Hint: $$\frac{1}{\sqrt{n+2}+\sqrt{n}}>\frac{1}{2\sqrt{n+2}}.$$

Answer 2

Alternatively, we have: $\displaystyle \sum_{n=1}^\infty \dfrac{1}{\sqrt{n+2} + \sqrt{n}} = \dfrac{1}{2}\displaystyle \sum_{n=1}^\infty \left(\sqrt{n+2}-\sqrt{n}\right) \geq \dfrac{1}{2}\cdot \displaystyle \lim_{n \to \infty} \sum_{k=1}^n \left(\sqrt{k+1}-\sqrt{k}\right) = \dfrac{1}{2}\cdot \displaystyle \lim_{n \to \infty} \left(\sqrt{n+1} - 1\right) = \infty$