TITLE: Determine : $\lim _{n \rightarrow \infty} \left[\frac{1 \cdot 1 !+2 \cdot 2 !+\cdots+n \cdot n !}{(n+1) !}\right]^{(n+1)!}$
QUESTION [12 upvotes]: What is the limit of the following :
$$
\lim _{n \rightarrow \infty} \left[\frac{1 \cdot 1 !+2 \cdot 2 !+\cdots+n \cdot n !}{(n+1) !}\right]^{(n+1)!}
$$
I think it is clear that the numerator approaches infinity faster than the denominator so the result should be $+\infty$?

REPLY [9 votes]: You have
$$a_n=\Bigg[\frac {\sum_{i=1}^n i\,i!}{(n+1)!}\Bigg]^{(n+1)!}=\Bigg[\frac {(n+1)!-1}{(n+1)!}\Bigg]^{(n+1)!}=\Bigg[1-\frac {1}{(n+1)!}\Bigg]^{(n+1)!} $$
$$\log(a_n)=(n+1)! \log\Bigg[1-\frac {1}{(n+1)!}\Bigg]\sim -1 $$
$$a_n \sim \frac 1e$$