## A divergent series …. or maybe not?

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,

## Iterated exponential integral

Prove that

**Solution**

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.

## Limit with zeta and Gamma function

Let denote the Riemann zeta function and denote the Euler’s Gamma function. Prove that

where stands for the Euler – Mascheroni constant.

**Solution**

Using the Laurent expansion of the function we have that

Using the Laurent expansion of the function we have that

The result now follows.

## On a log Gamma integral using Riemann sums

Evaluate the integral

using Riemann sums.

**Solution**

Partition the interval into subintervals of length . This produces,

(1)

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.