Convergence of Infinite Series $\sum\limits_{n=1}^{\infty} \frac{\sqrt{n+1}-\sqrt{n-1}}{n}$

Question

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

According to Mathematica, this series converges. I can't figure out how to prove this, however. If I split the series apart, I get two divergent series. The ratio test comes out to 1, which is inconclusive. Additionally, every series I can think of comparing this to is divergent (like 1/$\sqrt{n}$). Any ideas on how to prove that this converges?

Answer 1

Hint: Multiply both the denominator and numerator by $\sqrt{n-1} + \sqrt{n+1}$.