I am struggling to see a way of finding the infinite sum of the series

$\displaystyle 2sin(\frac{1}{k}-\frac{1}{k+1})cos(\frac{1}{k}+\frac{1}{k+1})$

$\displaystyle sin(\frac{1}{k}-\frac{1}{k+1})$ converges to 0 and $\displaystyle cos(\frac{1}{k}+\frac{1}{k+1})$ converges to 1.

Expanding and simplifying the trig functions just yields

$\displaystyle 2(sin(\frac{1}{k})cos(\frac{1}{k})cos(\frac{2}{k+1 })-sin(\frac{1}{k+1})cos(\frac{1}{k+1})cos(\frac{2}{k }))$

Which is not of any use

We didn't cover many methods for finding the sum of series in class, just ways of showing that a series converges

Could someone point me in the right direction?