3. Let the sequence $\left(x_{n}\right)_{n \in \mathbb{N}^{*}}$ be defined such that $x_{1}=1$ and $x_{n+1}=x_{n}+n$, for any $n \in \mathbb{N}^{*}$. Calculate $\lim _{n \rightarrow \infty} \sum_{k=2}^{n} \frac{1}{x_{k}-1}$.
3. $x_{n}=1+\frac{n(n-1)}{2}$ ..... $.3 p$
$\sum_{k=2}^{n} \frac{1}{x_{k}-1}=2\left(1-\frac{1}{n}\right)$ ..... $.3 p$
$\lim _{n \rightarrow \infty} \sum_{k=2}^{n} \frac{1}{x_{k}-1}=2$ ..... $1 p$