Monday, 24 November 2014

real analysis - Limit of bn when bn=fracann and limlimitsntoinfty(an+1an)=l




I have the following problem.



I have to find the limit of bn=ann, where limn(an+1an)=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+1an=(n+1)bn+1nbn



Assuming that bn converges to a real number m, we see that:



l=(n+1)mnm, 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+1an=an+1an(n+1)nl,n


where {n}nN is monotone and divergent, then
annl,n


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