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

analysis001
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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!
 
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analysis001 said:

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.
 
SammyS said:
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.
 
analysis001 said:
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 ?
 
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Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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