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?

Let the sum of series be denoted by $S$, the term corresponding to each k be $s_k$. Thus each $s_k$ is made of three terms.
$S=\sum s_k=\sum k(\frac{2}{k}-\frac{4}{k+1}+\frac{2}{k+2})$
$=\sum\limits_{k=1}^{\infty} (2-\frac{4k}{k+1}+\frac{2k}{k+2})$
Notice that last term of each $s_k$ and first 2 terms of $s_{k+1}$ have a sum $0$ as seen below:  $\frac{2k}{k+2}+2-\frac{4(k+1)}{k+2}=2 +\frac{-2k-4}{k+2}=0$ 
Taking first two terms of $s_0$ and last term of $s_n$ as $n$ goes to infinity:
$S=\lim_{n \to \infty} (2-\frac{4*1}{1+1}+\frac{2n}{n+2})=\lim_{n \to \infty} \frac{2n}{n+2}=2$ 
