I have the following problem.
I have to find the limit of bn=ann, where lim
My approach:
I express a_n in terms of b_n, i.e.
a_n=nb_n and a_{n+1}=(n+1)b_n
We look at the difference: a_{n+1}-a_n=(n+1)b_{n+1}-nb_n
Assuming that b_n converges to a real number m, we see that:
l=(n+1)m-nm, from where I conclude that m=l.
What I'm left with is proving that b_n is convergent which I'm not sure how to do.
Thanks in advance!
Answer
As @ParamanandSingh suggested, from Stolz–Cesàro theorem
a_{n+1}-a_n=\frac{a_{n+1}-a_n}{(n+1)-n} \rightarrow l, n \rightarrow \infty
where \{n\}_{n \in \mathbb{N}} is monotone and divergent, then
\frac{a_n}{n} \rightarrow l, n \rightarrow \infty
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