## Convergent sequence

Let be a sequence of positive real numbers such that

(1)

Prove that converges.

Solution

Fix and let . Then, there exist such that where . Thus,

Letting it follows that

Since this holds forall it follows that and the result follows.

## Root inequality

Let be positive real numbers satisfying the following equality

Prove that

Solution

We begin by stating a lemma:

Lemma: Let be positive real numbers, then:

Now, making use of the lemma we have that:

Making use of the Cauchy – Schwartz inequality we have that

The inequality now follows.

## Trigonometric identity

Let be positive real numbers such that . Prove the following trigonometric identities:

Solution

1. We have successively:

2. Similarly, we have successively:

## Number of solutions

Find the number of solutions of the equation:

in the positive integers.

Solution

We set . It suffices to find the number of solutions of the equation

(1)

in the non negative numbers. We represent each sum of non negative integers with a sequence of dots () followed by a vertical bar (), after dots another one vertical bar etc, till we place the last dots ( without the vertical bar at the end.) For example the sum can be represented as

We note that every solution of matches a sequence that has dots in total and vertical bars. Conversely, every such sequence matches a solution of .

Thus, in total there are

solutions.

Comment: In general the equation

has solutions in the positive integers and solutions in the non negative integers.

## Eulerian equality

We know that there are infinite Pythagorian triplets, that is numbers such that

(1)

Let us investigate if there exist triplets such that

(2)

where denotes the Euler’s totient function.

Solution

Indeed, there are infinite triplets such that is satisfied. For example noticing that

we deduce that for each natural such that we have