Convergence of series sqrt(n+1)-sqrt(n-1)

I need to determine whether the series converges or diverges using either the integral test, d'Alembert's ratio test or the comparison test.

So far, I can determine that the series diverges by finding a rearrangement using a finite series and then finding the limit as n approaches infinity:

Hence:

and so the series diverges.

However, I have not used either of the three tests mentioned. Is there a way to determine convergence/divergence using those tests?

I have ascertained that for higher powers of n (p>2) of the same series, the series would converge (using the comparison test):

since we know that this last series (a p-series) converges for and therefore since all the other series are lower in value and are positive series, they must also converge.

But for the case of as with this problem, the last series would diverge, and showing that a series is less than a diverging series proves nothing.

Thanks for your help! (Happy)