Finding limit of a (Laurent?) series I've been practicing series for my upcoming Calculus 1 exam, and I've stumbled upon this one:
$1 + \frac{1}{1 + 2} + \frac{1}{1 + 2 + 3} + ... + \frac{1}{1 + 2 + 3 + ... + n}$
The task is to find the limit.

As $1+2+\ldots +n = \frac{n}{2}(n+1)$ you are looking for the sum of the series 
$S = \sum_{n=1}^{\infty} \frac{1}{\frac{k}{2}(k+1)} = 2 \sum_{k=1}^{\infty} \frac{1}{k(k+1)}$.
Nos separating you get that $S_n = 2\sum_{k=1}^{n} \frac{1}{k} - \frac{1}{k+1} = 2(1 - \frac{1}{n+1})$. Now taking limit when $n \rightarrow \infty$ you get that $S=2$
