Convergence test of certain series: $\sum_{n=2}^{\infty}\frac{\sqrt{n+2}-\sqrt{n-2}}{\sqrt n}$

Question

I need to find out whether this sequence converges or diverges using limit comparison test.

$\sum_{n=2}^{\infty}\frac{\sqrt{n+2}-\sqrt{n-2}}{\sqrt n}$

I've tried it with the use of sequence $\frac{1}{\sqrt n}$ but it didn't help (the result was 0). Maybe I've done something wrong or there is other sequnce but I can't find it.

Answer 1

$\dfrac{\sqrt{n+2}-\sqrt{n-2}}{\sqrt n} = \dfrac{4}{\sqrt n(\sqrt{n+2}+\sqrt{n-2})}$

Compare it with $\sum \dfrac{1}{n}$

Answer 2

Hint: try multiplying the numerator and denominator by $\sqrt{n+2}+\sqrt{n-2}$.