<s>[INST] Why does this sum converge? I know that the following sum converges to 2 via WolframAlpha, but I am not sure why.  
$$\sum_{k=1}^\infty k \left[\frac{2}{k} - \frac{4}{k+1} + \frac{2}{k+2}\right] = 2$$
WolframAlpha gives the following partial sum formula:
$$\sum_{k=1}^n k \left[\frac{2}{k} - \frac{4}{k+1} + \frac{2}{k+2}\right] = \frac{2n}{n+2}$$
I would intuitively guess that the result of the partial sum formula is 2 for $n = \infty$. Where did that partial sum formula come from? Can someone help me build some intuition here?
 [/INST] $$\begin{align}k\left(\frac 2k-\frac{4}{k+1}+\frac{2}{k+2}\right)&=2-\frac{4k}{k+1}+\frac{2k}{k+2}\\&=2-4\left(1-\frac{1}{k+1}\right)+2\left(1-\frac{2}{k+2}\right)\\&=4\left(\frac{1}{k+1}-\frac{1}{k+2}\right)\end{align}$$
</s>