Determine : $\lim _{n \rightarrow \infty} \left[\frac{1 \cdot 1 !+2 \cdot 2 !+\cdots+n \cdot n !}{(n+1) !}\right]^{(n+1)!}$

[Note] : Notice that $n\cdot n! = (n+1)!-n!$, $\forall n\in\mathbb{N}$ go ahead and try to prove that using induction. It follows that : \begin{align*} \lim _{n \to \infty} \left[\displaystyle\frac{1 \cdot 1 !+2 \cdot 2 !+\ldots+n \cdot n !}{(n+1) !}\right]^{(n+1)!}&=\lim_{n\to\infty}\left[\frac{(n+1)!-1}{(n+1)!}\right]^{(n+1)!}\\ &=\lim_{n\to\infty}\left(1-\frac{1}{(n+1)!}\right)^{(n+1)!}\\ &=\boxed{\frac{1}{e}} \end{align*}

You have $$a_n=\Bigg[\frac {\sum_{i=1}^n i\,i!}{(n+1)!}\Bigg]^{(n+1)!}=\Bigg[\frac {(n+1)!-1}{(n+1)!}\Bigg]^{(n+1)!}=\Bigg[1-\frac {1}{(n+1)!}\Bigg]^{(n+1)!} $$ $$\log(a_n)=(n+1)! \log\Bigg[1-\frac {1}{(n+1)!}\Bigg]\sim -1 $$ $$a_n \sim \frac 1e$$