Re: interesting inequality
The question turned out to be easier than I thought .
Lnp is just the area under the curve from 1 to p of 1/x.
You can find an upper bound and a lower bound to express the approximate area as a summation .
The sum of all the upper rectangles from [1,2] ,[2,3],..[p-1,p] is just 1+1/2+1/3+... 1/p
The sum of all the lower rectangles is 1/2+1/3+...1/p .
Lnp must be greater than the sum of all the lower rectangles and lower than the sum of all the upper rectangles .
For the second problem , just use a trapezoid for the intervals [1,2],[2,3],[3,4]...[p-1,p] and show that lnp is less than the sum of all the trapezoids but greater than the sum of all the lower rectangles .
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