Say what? This one blew my mind when I first encountered it. But it turns out Euler was the one who came up with it and it’s proof is just beautiful!
Say you have a quadratic equation whose roots are , then you can write as follows:
As for as this proof is concerned we are only worried about the coefficient of , which you can prove that for a n-degree polynomial is:
where are the n-roots of the polynomial.
Now begins the proof
It was known to Euler that
But this could also be written in terms of the roots of the equation as:
Now what are the roots of ?. Well, when i.e *
The roots of the equation are
Comparing the coefficient of y on both sides of the equation we get that:
* n=0 is not a root since
at y = 0
** Now if all that made sense but you are still thinking : Why on earth did Euler use this particular form of the polynomial for this problem, read the first three pages of this article. (It has to do with convergence)