## Parametric improper integral

Let such that . Prove that

## Improper Gamma integral

Let denote the Euler’s Gamma function. Prove that

where .

## Arithmotheoretic sum

Evaluate the limit:

Solution

However,

Hence

## Series of Bessel function

Let denote the Bessel function of the first kind. Prove that

Solution

The Jacobi – Anger expansion tells us that

(1)

Hence by Parseval’s Theorem it follows that

## Proof of “Fermat’s last theorem”

Let and . Prove that the equation

has no solution.

Solution

Without loss of generality , assume that . If held , then it would be thus . It follows from Bernoulli’s inequality that,

which is an obscurity. The result follows.

### Who is Tolaso?

Find out more at his Encyclopedia Page.