<s>[INST] The sum of infinite series $\sum_{k=1}^{\infty}2\sin\left(\frac{1}{k}-\frac{1}{k+1}\right)\cos\left(\frac{1}{k}+\frac{1}{k+1}\right)$ 
Determine the sum of the  infinite series  
$$\sum_{k=1}^{\infty}2\sin\left(\frac{1}{k}-\frac{1}{k+1}\right)\cos\left(\frac{1}{k}+\frac{1}{k+1}\right).$$

 [/INST] This turns out to be a telescoping sum:
$$\sum_{k=1}^{\infty} \left ( \sin{\frac{2}{k}} - \sin{\frac{2}{k+1}}\right ) = \sin{2} - \sin{1} + \sin{1} - \sin{\frac{2}{3}} + \ldots = \sin{2}$$
This sum converges because, as $k \rightarrow \infty$, the summand approaches
$$\frac{2}{k (k+1)} \sim \frac{2}{k^2}$$
so the sum converges by the comparison test.
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