The number ranges over all possible powers with both the base and the exponent positive integers greater than , assuming each such value only once. Prove that:
Let us denote by the set of positive integers greater than that are not perfect powers ( i.e are not of the form , where is a positive integer and ). Since the terms of the series are positive , we can freely permute them. Thus,
Let be a Lipschitz function . Prove that forall it holds that
We have that
We begin by stating a lemma:
Lemma: Let be an analytic function on some closed disk which has center and radius . Let denote the the boundary of the disk. It holds that
Proof: By the Cauchy integral formula we have that
The equation of a circle of radius and centre is given by . Hence,
and the proof is complete.
Something quickie: Given the assumptions in Gauss’ MVT, we have
The proof of the result is pretty straight forward by using the fact that
Back to the problem the result now follows by the lemma.
Evaluate the integral
using Riemann sums.
Partition the interval into subintervals of length . This produces,
On the other hand, assuming is even:
since it holds that
Euler’s Gamma reflection formula was used at line . Letting we get that
If is odd we work similarly.