To restate the problem:
Using , and
show that for , we have
Is that correct?
If so, I would compute the integral
and combine all the terms into one logarithm.
That should give you some direction.
Using f(x) = lnx, and f(n-1) < Sn n-1 f(x) dx < f(n)
Show that for n>1
(n-1)! < n^n (n-1)! / (e(n-1)^(n-1)) < n!
Have tried to approach this using graphs, but could not get middle thing. Done this by adding the rectangle below f(x) and rectangles above f(x) with lengths of 1 unit square, starting with x =1