## 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

## A limit!

Evaluate the limit:

Solution

Let . Then,

It follows by Stolz–Cesàro that

Hence .

## 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:

### Who is Tolaso?

Find out more at his Encyclopedia Page.