Determine whether series is convergent or divergent

[Nan1teZ]

1. The problem statement, all variables and given/known data

Determine whether or not the series \sum^{\infty}_{n=1} \frac{1}{\sqrt{n+1}+\sqrt{n}} converges.

3. The attempt at a solution

Assuming this diverges, I rationalize it to get get \sum^{\infty}_{n=1} \sqrt{n+1} - \sqrt{n}. How would I proceed further?

Is this even the right approach?

[snipez90]

Well, you can't just assume it diverges. I haven't studied series in quite a while. The rationalizing of the denominators seems like a promising approach, but someone else will have to comment on that.

I think the following is a valid method. Let f(n) be the expression we are summing. As n-> +inf f(n) -> 1/(2sqrt(n)). But the infinite series with 1/sqrt(n) is a p-series and I think it's divergent.

[Dick]

snipez90 is right. Don't 'rationalize' it. Just compare it with a divergent p-series.