Friday, 15 December 2017

sequences and series - Prove the inequality for all N



Show that the following inequality holds for all integers N1
|Nn=11n2Nc1|c2N
where c1,c2 are some constants.



I have tried induction but it doesn't seem promising.
Any ideas please?



Answer



Since 1/n is decreasing, we can get an upper bound on the sum by computing: 1+N11xdx=1+2N



Thus we have Nn=11n2N=(1+2N)2N+E(N)=1+E(N)



Where E(N)=1+N11xdxNn=11n>0



is the error of our estimation at the point N. Now can you estimate E(N)?







Using the formula from Corollary 2.4 in the pdf linked in the comments, we find that Nn=11n=N11xdxN1{x}1/2(x)3/2dx+1



Here {x} is the fractional part of x, which is always less than 1. The middle integral can be estimated by |N1{x}1/2(x)3/2dx|<N111/2(x)3/2dx=11N


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