Calculate $\lim_{n\to\infty} (\frac{1}{{1\cdot2}} + \frac{1}{{2\cdot3}} + \frac{1}{{3\cdot4}} + \cdots + \frac{1}{{n(n + 1)}})$ 
Calculate

$$\lim_{n\to\infty} \left(\frac{1}{1\cdot2} + \frac{1}{2\cdot3} + \frac{1}{3\cdot4} +  \cdots + \frac{1}{n(n + 1)}\right).
$$
If reduce to a common denominator we get
$$\lim _{n\to\infty}\left(\frac{X}{{n!(n + 1)}}\right).$$
How can I find $X$ and calculate limit?

This won't get you far. Try $\frac{1}{n(n + 1)} = \frac{n + 1 - n}{n(n+1)} = \frac{1}{n} - \frac{1}{n + 1}$ instead.
