
Sum of the series
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?

Re: Sum of the series
$\displaystyle 2\sin\left(\frac1k  \frac1{k+1}\right)\cos\left(\frac1k + \frac1{k+1}\right)\,=\,\sin\frac2k\sin\frac2{k+1}$
$\displaystyle \sum_{k=1}^n\left[\sin\frac2k\sin\frac2{k+1}\right]\,=\,\sin2\sin\frac2{n+1}\to\sin2\ \mbox{as}\ n\to\infty$

Re: Sum of the series
At first I didnt see what you did but now I see its a telescoping series! thank you