How to compute $ \lim_{n\rightarrow\infty}{\frac{1+2\cdot2!+\dots+n\cdot n!}{(n+1)!}}$?

Question

I'm asked to find the limit of the following sequence:

$$ \lim_{n\rightarrow\infty}{\frac{1+2\cdot2!+\dots+n\cdot n!}{(n+1)!}}$$

I've tried using the Stolz Theorem with $a_n=1+2\cdot2!+\dots+n\cdot n!$ and $b_n=(n+1)!$. As $\frac{a_{n+1}-a_n}{b_{n+1}-b_n} \rightarrow1$ I concluded that the limit of the sequence is 1.

I wanna know if my result is correct and if there is another way to find this limit.

Answer 1

HINT

You can make use of the following identity, which can be proved through induction: \begin{align*} 1 + 2\cdot 2! + 3\cdot 3! + \ldots + n\cdot n! = (n+1)! - 1 \end{align*}