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.