## A log – trigonometric integral

Prove that

**Solution**

We state two lemmata:

**Lemma 1: **

*Proof:*

Successively we have:

where is the Dirichlet eta function and the Riemann zeta function.

**Lemma 2: **It holds that .

*Proof:*

Successively we have:

where is the Dirichlet Beta function.

Hence,

## MacLaurin of exp(exp(x))

Let denote the -th Bell number. Prove that

**Solution**

Taking derivatives we get that

where and are the Stirling numbers of second kind. We also note that

where are the Bell numbers. Thus,

## A root limit

Let be positive real numbers such that . Prove that

**Solution**

Without loss of generation , let . Then,

since forall . Thus, by the squeeze theorem it follows that

## Floor series

Let denote the floor function. Evaluate the series

**Solution**

First of all we note that and are never squares. Thus, there exists a positive integer such that

It is easy to see that and thus we conclude that

Now is equal to the even number if-f

Hence, since the series is absolutely convergent we can rearrange the terms and by noting that the finite sums are telescopic , we get that: