## ζ(3) tail in floor function

Let . Prove that

**Solution**

We note that

as well as

The result follows.

**Note: **It also holds that

Using the same exact technique we see that

as well as

The result follows.

## Multiple logarithmic integral

Prove that

**Solution**

First of all we observe that the integral as well as the integral diverge whereas the proposed integral converges which is an interesting fact. Now,

since is determined by the inequalities

## A beautiful Gamma series

Let denote the Gamma function. Prove that

**Solution**

The Taylor series is

Hence,

and the result follows.

## On a limit with summation

Let . Prove that

**Solution**

Let us begin with the simple observation that

Now, here comes a handy lemma.

**Lemma: **Let be a differentiable function with continuous derivative. It holds that

*Proof: *The derivation of the theorem follows from application of MVT in the interval .

Hence the limit follows to be .

**Note: **Applying Euler – MacLaurin we have

## Limit of a multiple integral

Prove that

**Solution**

Let be independent and uniform random variables. By the law of large numbers we have

in probability as . Therefore ,

in probability as .

The ratio random variables are bounded below by zero and above by one. This guarantees convergence of the expectations, as well.

So,

which is the required result.

**Remark: **In general it holds that

because the distribution of is the Lebesgue measure on hence for every measurable function ,