# Investigate the convergence or divergence properties of $\sum_{n=1}^{\infty}a_n$.

Investigate the convergence or divergence properties of $\sum_{n=1}^{\infty}a_n$, where $a_n = \sqrt{n+1} - \sqrt{n}$.

I multiplied by its complex conjugate and resulted in $\frac{1}{\sqrt{n+1} + \sqrt{n}}.$ Then I tried applying the ratio test but that was inconclusive. I don't know how to apply the root test.

Neither the root nor the ratio test will work. You do have this $$\sqrt{n+1} - \sqrt{n} = {1\over \sqrt{n+1} + \sqrt{n} } \sim {1\over 2\sqrt{n}}.$$ How does $\sum_n 1/\sqrt{n}$ behave?
• I can't conclude from this, it just gives me a ratio test of $1$ again. – Don Larynx Dec 1 '13 at 0:32
• @DonLarynx What is the definition of convergence of a series? Here we are looking at the sum of the first $N$ terms. – user17762 Dec 1 '13 at 0:32