How can I represent this as a sum? I am solving probability, and I got the need to know what this sum is:
$ \frac{1}{2} \times \frac{1}{3} + \frac{1}{2}\times \frac{2}{3} \times \frac{1}{4} + \frac{1}{2}  \times \frac{2}{3} \times \frac{3}{4} \times \frac{1}{5} + \cdots$
I tried much with no success. How can I approach such sums during an exam?      

In this case, it is useful to simplify somewhat. We get 
$$\frac{1}{2\cdot 3}+\frac{1}{3\cdot 4}+\frac{1}{4\cdot 5} + \frac{1}{5\cdot 6}+\cdots.$$
This is 
$$\left(\frac{1}{2}-\frac{1}{3}\right)+\left(\frac{1}{3}-\frac{1}{4}\right)+\left(\frac{1}{4}-\frac{1}{5}\right)+ \left(\frac{1}{5}-\frac{1}{6}\right)+\cdots.$$
Note the telescoping (almost everything cancels).
Remark: An expression for the sum is $\sum_{1}^{\infty} \frac{k!}{(k+2)!}$.
Telescoping is an idea that comes up moderately often, so it is worth watching out for. On exams, the most common closed from things to look out for a geometric series, the expansion of $e^x$ for some $x$, and telescoping series. 
