## Non linear system

Solve the system

Solution

We set and where . The second equation of course is written as

Now ‘s follow.

Historical note: In the Babylonian signs , tracing back in 1700 BC , there are a lot of geometrical problems that are equivalent to the solution of such systems. In order to be solved the following ( modern ) formulae were used

The above procedure actually led to the discriminant in order for us to solve a second order equation. For example in a book of that age we see the equation . Try to solve this using the above technique.

The above exercise , along with the historical note , can be found at mathematica.gr .

## The series converges

Prove that the series

converges. Examine if the convergence is absolute.

Solution

Lemma: Let such that then the sequence

is bounded.

With the above lemma in mind the series converges as a consequence of Dirichlet’s theorem. The fact that the series does not converge absolutely can be seen by applying the Jordan inequality

(1)

that

hence the series diverges absolutely.

## Range of convergence

For what values of does the series

converge?

For none. Since for the sequence it holds that we deduce that the potential limits are or . Hence, the sequence cannot tend to zero and the conclusion follows.

Note: One interesting question is the following. For what values of does the series

## Irreducible fraction

Find all positive integers such that it holds that

(1)

where stands for the period.

Solution

Well,

## On the supremum and infimum of a sine sequence

Let be a sequence defined as

Find the supremum as well as the infimum of the sequence .

Solution

Background: This problem was on the shortlist of the 2014 Olimpiada Nationala de Matematica de Romania and was suggested by Leo Giugiuc.

We begin by the very well known manipulation.

Thus and we have to find the supremum and infimum of . Since the values are dense on the unit circle , the same shall hold for implying that and . Thus,