Let be a sequence of positive real numbers such that
Prove that converges.
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.
Let be positive real numbers satisfying the following equality
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.
Let be positive real numbers such that . Prove the following trigonometric identities:
- We have successively:
- Similarly, we have successively:
Find the number of solutions of the equation:
in the positive integers.
We set . It suffices to find the number of solutions of the equation
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
Comment: In general the equation
has solutions in the positive integers and solutions in the non negative integers.
We know that there are infinite Pythagorian triplets, that is numbers such that
Let us investigate if there exist triplets such that
where denotes the Euler’s totient function.
Indeed, there are infinite triplets such that is satisfied. For example noticing that
we deduce that for each natural such that we have