Why is this sum what it is? I am trying to prove that
$$\frac{1}{1} + \frac{1}{1+2} + \frac{1}{1+2+3} + \cdots + \frac{1}{1+2+3+\cdots+n} = \frac{2n}{n+1}$$
I know the denominators have a closed form $\sum_{i=1}^{k} i = \frac{k(k+1)}{2}$ implying 
$$\frac{1}{1} + \frac{1}{1+2} + \frac{1}{1+2+3} + \cdots + \frac{1}{1+2+3+\cdots+n} = 2\sum_{k=1}^{n}\frac{1}{k(k+1)}$$
Can't figure out where to go from here.

Hint Notice that : $$\frac{1}{k(k+1)}=\frac{1}{k}-\frac{1}{k+1}.$$
