# 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:

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

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