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## Arithmotheoretic sum

Evaluate the limit:

**Solution**

However,

Hence

## Series of Bessel function

## 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.

## Limit of a sequence

Let be a continuous function and be the set of all positive integers such that there exists such that

Prove that is infinite and evaluate the limit

## A factorial limit

Let denote the factorial of a real number; that is . Evaluate the limit:

**Solution**

It holds that

where denotes the -th harmonic number and the Euler – Mascheroni constant.