Friday, 7 April 2017

real analysis - If sumlimitsin=1nftynan converges , does sumlimitsin=1nftynan+1 converge?



I ask for some help with this question:



Prove or provide counter example:



If n=1nan converges then n=1nan+1 also converges.




I tries this way:



If n=1nan converges then nan0, therefore an0.



There are 3 possible cases:



1) If an>0 and an is monotonic decreasing sequence then $na_{n+1}

2) If an>0 and an is not monotonic decreasing sequence : it is not possible that an+1>an because in this case an, therefore it must be an+1an and n=1nan+1 converges by Comparison Test.




3) If an is sign-alternating series. There I have a problem to find a solution.



Thanks.


Answer



Yes. Put bn=nan, so the question is now (see my comment on the question):




If n=1bn converges, does n=1bnn converge?





Let sn=nk=1bn. We get (partial summation)
nk=1bkk=nk=1sksk1k=nk=1(1k1k+1)sk+snn+1=nk=11k(k+1)sk+snn+1


which converges as n, because sk is bounded, so the sum is absolutely convergent.


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