Cauchy's root test for series divergence Just a question regarding determining the divergent in this example :$$\sum{ 1 \over \sqrt {n(n+1)}} $$ is divergent. It explains the reason by saying that $a_n$ > $1 \over n+1$. If I am not wrong it uses the root test but should not we have the $a_n \ge 1$ but how does their reasoning assure us this exactly?

Or still simpler, with equivalences, since it is a series with positive terms:
$$\frac1{\sqrt{n(n+1)}}\sim_\infty\frac 1n,$$
which diverges, hence the original series diverges.
