I'm having problems with following limit:
This is what I did now: I already proved this sequence is monotone descending, at least for $\;n\ge4\;$ , and since zero , or even one, is a low bound limit exists.
Also think limit is $\;1\;$ since $\;n!\;$ is $\;n\;$ times greater than $\;(n-1)!\;$ , so in numerator we can like omit all summands except last one, but of course I need something formalizing more.
I don't know of Stirling approximations or integrals, but someone told me that won't help in this.
Any help is appreciated.