<s>[INST] 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.
 [/INST] Hint. On may observe that
$$
\frac{1}{n(n+1)(n+2)}=\frac12\frac{2}{n(n+1)(n+2)}=\frac12\frac{(n+2)-n}{n(n+1)(n+2)}
$$ giving
$$
\frac{1}{n(n+1)(n+2)}=\frac{1}{2n(n+1)}-\frac{1}{2(n+1)(n+2)}
$$ the given sum is thus a telescoping one.
Hope you can finish it.
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