## Zeta function reciprocal limit

Evaluate the limit

Solution

We note that

Hence

Thus . The limit follows to be .

Note: It holds that .

## 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