Calculate

$$
\frac{1}{1 \times 2}+\frac{1}{2 \times 3}+\frac{1}{3 \times 4}+\ldots+\frac{1}{2013 \times 2014}
$$

## - Inequalities -

For common inequalities, see the handouts from previous sessions, as well as the Animath course written by Pierre Bornsztein.
The trick is to notice that for any strictly positive integer $k$, $\frac{1}{k(k+1)}=\frac{k+1-k}{k(k+1)}=\frac{1}{k}-\frac{1}{k+1}$. By telescoping, the desired sum is therefore $1-\frac{1}{2014}=\frac{2013}{2014}$.