Without using L'Hospital's rule, I want to find a limit of the following. 
Given a series with $a_n = \sqrt{n+1}-\sqrt n$ , determine whether it converges or diverges.

The ratio test was inconclusive because the limit equaled 1.
So I tried to use the root test. So the problem was reduced to finding $$\lim_{n \to \infty} \sqrt[n]{\sqrt{n+1}-\sqrt n}$$
Since the book hasn't covered derivatives, yet, I am trying to solve this without using L'Hospital's rule. So the following is the strategy that I am trying to proceed with.
I cannot come up with a way to simplify the expression, so I am trying to compare $a_n$ to a sequence that converges to 1 or maybe something less. In fact, I want to compare it to 1 because of that reason.
The problem with this is that adding constants makes the series divergent, and I suspect that the series converges, so I don't think that works at all.
Can someone help ?

$$a_n = \sqrt{n+1}-\sqrt n=\dfrac{1}{\sqrt{n+1}+\sqrt{n}}\ge\dfrac{1}{2\sqrt{n+1}}=b_n~(say)\\\displaystyle\sum b_n\text{ is divergent by p series test and so is } \displaystyle\sum a_n.$$
