I have the following problem.
I have to find the limit of bn=ann, where limn→∞(an+1−an)=l
My approach:
I express an in terms of bn, i.e.
an=nbn and an+1=(n+1)bn
We look at the difference: an+1−an=(n+1)bn+1−nbn
Assuming that bn 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 bn is convergent which I'm not sure how to do.
Thanks in advance!
Answer
As @ParamanandSingh suggested, from Stolz–Cesàro theorem
an+1−an=an+1−an(n+1)−n→l,n→∞
where {n}n∈N is monotone and divergent, then
ann→l,n→∞
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