integration – Where is $ e $ hidden in this sequence?

To let $ a_n $ a sequence that is recursively defined as:

$ a_0 = c-1 space $ $ (c in mathbb {R}) $

$ a_n = na_ {n-1} -1 $

Alternative, $ a_n $ can be defined as:

$ a_n = cn! – int_1 ^ infty x ^ ne ^ {1-x} dx $

Now we have to show that:

$ lim_ {n rightarrow infty} a_n = begin {cases} – infty & text {for $ ce $} end {cases} $

…With $ e $ Be Euler's number. I really did not expect it $ e $ To hide in that order, alright, I think math likes to be surprising. But where do I find that? I tried to play a bit with the integral, and maybe somehow to come to it $ e $Expression, but unfortunately I could not find anything. Could you maybe give me a hint? Thank you in advance!

Oh, and there are tips (that need not be proven) for this task:

$ n! = int_0 ^ infty x ^ ne ^ {- x} dx $

$ lim_ {n rightarrow infty} int_0 ^ 1x ^ ne ^ {- x} dx = int_0 ^ 1 lim_ {n rightarrow infty} x ^ ne ^ {- x} dx $