Show that the following inequality holds for all integers N≥1
|N∑n=11√n−2√N−c1|≤c2√N
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+∫N11√xdx=1+2√N
Thus we have N∑n=11√n−2√N=(1+2√N)−2√N+E(N)=1+E(N)
Where E(N)=1+∫N11√xdx−N∑n=11√n>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 N∑n=11√n=∫N11√xdx−∫N1{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|<∫N11⋅1/2(x)3/2dx=1−1√N
No comments:
Post a Comment