Consider the sequence defined recursively as
Prove that .
Lemma: If is a sequence for which then
Proof: In Stolz theorem we set and .
It is easy to see that is is monotonically decreasing to zero. Moreover, an application of L’Hospital’s rule gives
Now, due to the lemma we have and the result follows.
Remark : The asymptotic now follows to be .
Problem: Find what inequality should satisfy such that the series