Proving Series Convergence: \sum_{n=1}^{\infty}\frac{\sqrt{n+1}-\sqrt{n}}{n}

[analysis001]

Homework Statement

Prove that the series \sum_{n=1}^{\infty}\frac{\sqrt{n+1}-\sqrt{n}}{n} converges.

The Attempt at a Solution

I think I'm going to use the comparison test but I'm having trouble coming up with a series to compare it to. Any clues would be great. Thanks!

 

[SammyS]

Homework Statement

Prove that the series \sum_{n=1}^{\infty}\frac{\sqrt{n+1}-\sqrt{n}}{n} converges.

The Attempt at a Solution

I think I'm going to use the comparison test but I'm having trouble coming up with a series to compare it to. Any clues would be great. Thanks!

Try rationalizing the numerator.

 

[analysis001]

Try rationalizing the numerator.

Yeah, I've gotten to that point, so as of now I have: \sum_{n=1}^{\infty}\frac{1}{n(\sqrt{n+1}+\sqrt{n})} but I'm still not sure what to compare it to.

 

[SammyS]

Yeah, I've gotten to that point, so as of now I have: \sum_{n=1}^{\infty}\frac{1}{n(\sqrt{n+1}+\sqrt{n})} but I'm still not sure what to compare it to.

Let's see ...

$\sqrt{n+1}\ \ > \sqrt{n}$

How can that help ?