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## 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.

## 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.

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