Limit of a sequence : $x_n = \frac{1}{1\cdot 2\cdot 3} + \frac{1}{2\cdot 3\cdot 4} + \ldots + \frac{1}{n(n+1)(n+2)}$ Can someone help me with this problem?
Finding the limit $\lim_{n \to \infty}\ x_n$ where
$$x_n = \frac{1}{1\cdot 2\cdot 3} + \frac{1}{2\cdot 3\cdot 4} + \ldots + \frac{1}{n(n+1)(n+2)},\quad n\in\mathbb{N}.$$
I don't have a clue how to do this.

From the question here we find that
$$\sum_{r=1}^\infty \frac 1{r^\overline{m}}=\frac 1{(m-1)(m-1)!}$$
Putting $m=3$ gives
$$\sum_{r=1}^\infty \frac 1{r^\overline{3}}=\sum_{r=1}^\infty \frac 1{r(r+1)(r+2)}=\frac 1{2\cdot 2!}=\color{red}{\frac 14}$$
