Thursday, 14 April 2016

real analysis - How to prove this sequence is convergent?






Suppose that the series k=1ak converges. Prove that
limn1nnk=1kak=0





I tried to use the definition of convergence of ak but I'm struggling to find N such that given ϵ>0, if n>N, then ϵ<1nnk=1kak<ϵ but I don't know how to connect them



hope somebody can help


Answer




Let sn=a1++ana. Then
nk=1kak=nk=1k(sksk1)=nk=1kskn1k=1(k+1)sk=nsnn1k=1sk.


Hence
1nnk=1kak=snn1n1n1n1k=1skaa=0.

We have used the fact that: If bnb, then so does b1++bnn.




Note. However, it is not in general true that nan0.


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