Euler’s crazy sums

Rigour is the hygiene of the mathematician, but it is not its source of nutrients… Here you have a fantastic piece of work by Leonhard Euler, where he showed how to reason mathematically, maybe without rigour, but with a rich and healthy intuition.

In the early days of calculus, people were fascinated by power series. Euler knew very well the Taylor series of the sine around zero:

\sin(x)=x-{x^3\over 3!}+{x^5\over 5!}-{x^7\over 7!}+\cdots

OK, so the sine function, in a sense, is a polynomial… Well… We know a few things about polynomials, don’t we? For example. If P(x) is a degree 3 polynomial and its roots are x_0, x_1 and x_2, then:

P(x) = K (x-x_0)(x-x_1)(x-x_2)

And the constant can be fixed if we know a single value, for example, P(0). So… let us do it with the sine function. We know its zeroes: n\pi for all integer n:

\sin(x) = K \cdots (x+3\pi) (x+2\pi) (x+\pi) x (x-\pi) (x-2\pi) (x-3\pi) \cdots

A little bit of regrouping:

{\sin(x)\over x} = K (x^2-\pi^2) (x^2-(2\pi)^2) (x^2 - (3\pi)^2) \cdots

Now, what can the constant be? If x=0, then \sin(x)/x is 1. A trick and we get rid of the K:

{\sin(x)\over x} = \left( 1 - {x^2\over \pi^2}\right) \left( 1 - {x^2\over (2\pi)^2} \right) \cdots

Waw, a remarkable formula on its own! You can check it numerically: it converges slowly beyond the first maximum, but it converges indeed. But we can get more from it. It should coincide with the Taylor series of the sine, right? (hm…) The constant term is easy, just 1. The quadratic term is not too hard either. Just add up all the products which consist of all ones and a single x^2 term. You get:

-{x^2\over \pi^2} - {x^2\over (2\pi)^2} - {x^2\over (3\pi)^2} - \cdots = -{x^2\over 3!}

to make it equal to the Taylor term for x^2. Now, make the coefficients equal:

-\sum_{n=1}^\infty {1\over n^2 \pi^2} = -{1\over 3!}

Or, equivalently,

\sum_{n=1}^\infty {1\over n^2} = {\pi^2\over 6}

Waw! Of course, there are rigorous ways to prove this formula. The first one I learnt was using Fourier series. But this one is funny, isn’t it? :)

Not all of Euler’s crazy sums were right. Some were amazingly wrong. But the wrongs of the genius are also funny… So I will soon discuss them here.

Happy summing!