Consider the sequence
and define,
.
n,k positive natural numbers.
Find:
Using induction I found that
But I don't know how to find the limit.
The limit should be
I have been taught to avoid L'Hospital' rule, so here is what I wrote:
and that's it.
Moreover, the same method works for this general problem. Let us write for simplicity, and , for all .
Let . Start from:
.
The definition of is just an intricate way to describe the coefficients in this expansion.
Indeed,
,hence ,
,hence ,
and so on, "unfolding the expansion", until
,hence .
If you really want to write every word of the proof, then you have to explicitate the above induction. Let and prove by induction on that :
(hence ).
(you could also use convergent series instead of asymptotic expansions to shorten the proof, but these would be a little inappropriate here since we only care for the limit)