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Sequence limit recursive formula

2175ExpertCalculusSequences & Series

OLIMPIADA DE MATEMATICĂ ETAPA LOCALĂ

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