Question 149
Consider the sequence $\left(x_{n}\right)_{n \geq 1}$ with $x_{1}=\frac{1}{3}$ and $x_{n+1}=(n+1) x_{n}+\frac{(n+1)(n+1)!}{3^{n+1}}, \forall n \in \mathbb{N}^{*}$.
Compute $\lim _{n \rightarrow \infty} n\left(\frac{x_{n}}{n!}-\frac{3}{4}\right)$.
Compute $\lim _{n \rightarrow \infty} n\left(\frac{x_{n}}{n!}-\frac{3}{4}\right)$.