Tuesday, 30 April 2019

calculus - True/False: mathoplimlimitsntoinftyanoverbn=1 implies suman,sumbn converge or diverge together.



limnanbn=1
Prove the statement implies an,bn converge or diverge together.
My guess the statement is true.



if an diverges, then limnan0



So,
limnan=L0limnanlimnbn=1Llimnbn=1L=limnbn0



therefore,
bn also diverges.



What I was not managed to do is proving that the two series converges together.
Or maybe the statement is not always true?


Answer



Surprisingly, this statement is false. For a simple counter-example, consider
an=(1)nn,andbn=(1)nn+1n
The condition anbn holds but an is convergent whereas bn is divergent.


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