Evaluate the limit

**Solution**

We note that

Hence

Thus . The limit follows to be .

**Note: **It holds that .

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# Joy of Mathematics

## Zeta function reciprocal limit

## ζ(3) tail in floor function

## Multiple logarithmic integral

## A beautiful Gamma series

## On a limit with summation

A site of university mathematics

Evaluate the limit

**Solution**

We note that

Hence

Thus . The limit follows to be .

**Note: **It holds that .

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.

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

Let denote the Gamma function. Prove that

**Solution**

The Taylor series is

Hence,

and the result follows.

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