TITLE: Does $\sum_{n=1}^{\infty} (\frac{1}{n}-\frac{1}{\sqrt{n(n+1)}})$ converge
QUESTION [3 upvotes]: $$\sum_{n=1}^{\infty} \left ( \frac{1}{n}-\frac{1}{\sqrt{n(n+1)}}\right )=\sum_{n=1}^{\infty} \left ( \frac{\sqrt{n(n+1)}-n}{n\sqrt{n(n+1)}}\right )$$
By ratio test I get $$ \lim_{n\to \infty} \frac{\frac{\sqrt{n(n+1)}-n}{n\sqrt{n(n+1)}}}{\frac{1}{n}}=\frac{0}{0}$$
So i can't conclude anything. I tried comparison tests, but i get $\le$ something divergent.

REPLY [0 votes]: You don't need ratio test or a clever bound for the general term... Like most problems on MSE, a simple asymptotic expansion fits the bill.
$$\frac{1}{n}-\frac{1}{\sqrt{n(n+1)}} = \frac{1}{2n^2} + o\left(\frac{1}{n^2}\right)$$
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