Evaluate the series:

*(Putnam Competition , 2016)*

**Solution**

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

Skip to content
# Joy of Mathematics

## Double series

## On an entire function

## A trigonometric identity

## A rational number

## An inequality

A site of university mathematics

Evaluate the series:

*(Putnam Competition , 2016)*

**Solution**

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

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.

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:

Let . Prove that the number

is rational.

**Solution**

We have successively that

since every number has an expansion of the form

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: