Double series

Evaluate the series:

(Putnam Competition , 2016)

Solution

Since the double series the series converges absolutely we can interchange the summation. Thus:

On an entire function

Let and be an entire function. Prove that for any arbitrary positive numbers it holds that:

Solution

Since our function is entire this means that it is holomorphic and can be represented in the form

This series converges uniformly on thus we can interchange summation and integral. Hence:

where is Kronecker’s delta. Similarly for the denominator. Dividing we get the result.

A trigonometric identity

Prove that

Solution

It suffices to evaluate the value of

Indeed:

The principal value of the quantity we are seeking lies in the interval . Thus and consequently we get the result.

Thus:

Note: In general , for positive it holds that:

A rational number

Let . Prove that the number

is rational.

Solution

We have successively that

since every number has an expansion of the form

An inequality

Let be positive real numbers such that

Prove that

(Vojtech Jarnik / Second Category / 2016)

Solution

Using the AM – GM inequality we have that

as well as . Thus: