Mishandling polynomials for fun and profit

One of the more surprising results a mathematician comes across in a university course is that the infinite sum $S = 1 + \frac{1}{4} + \frac{1}{9} + ... + \frac{1}{n^2} + ...$ comes out as $\frac{\pi^2}{6}$. If $\pi^2$s are going to crop up in sums like that, they should be required to explain themselves!

There is, though, a lovely - if not strictly rigorous - argument for why it should be. (It was put forward by Euler, which is a pretty good argument from authority, if nothing else.)

Consider the function $f(x)=\frac{\sin(x)}{x}$. Anyone with a little bit of analysis under their belt knows that $f(x)$ approaches 1 as $x$ approaches 0; it's also pretty clear that it has zeros at $\pm \pi$, $\pm 2\pi$, and generally at $\pm n\pi$ for natural $n$.

Factorising!

If we were to approximate it as a polynomial, we might write it in factorised form as $f_F(x) =
\left(1-\frac{x}{\pi}\right)
\left(1+\frac{x}{\pi}\right)
\left(1-\frac{x}{2\pi}\right)
\left(1+\frac{x}{2\pi}\right)
\left(1-\frac{x}{3\pi}\right)
\left(1+\frac{x}{3\pi}\right)...
\left(1-\frac{x}{n\pi}\right)
\left(1+\frac{x}{n\pi}\right)...$. Better still, spotting that each pair of brackets is the difference of two squares, we can instead go for $f_F(x)=
\left(1-\frac{x^2}{\pi^2}\right)
\left(1-\frac{x^2}{4\pi^2}\right)
\left(1-\frac{x^2}{9\pi^2}\right)
...
\left(1-\frac{x^2}{n^2\pi^2}\right)
...$.

That's all very pretty, and has some $\pi^2$s in but what does it have to do with the price of fish?

Maclaurin!

Well, you see, $f(x)$ also has a Maclaurin series: $f_M(x) = 1 - \frac{x^2}{6} + \frac{x^4}{120} - ...$

If we were to expand $f_F(x)$ up to the $x^2$ term, we'd get $1 - \frac{x^2}{\pi^2} - \frac{x^2}{4\pi^2} - \frac{x^2}{9\pi^2} - ... - \frac{x^2}{n^2 \pi^2} - ...$.

The price of fish!

Matching coefficients for the $x^2$ term gives $\frac{1}{6} = \frac{1}{\pi^2} + \frac{1}{4\pi^2} + \frac{1}{9\pi^2} + ... + \frac{1}{n^2 \pi^2} + ...$.

Multiplying both sides by $\pi^2$ gives the result we're after: $\frac{\pi^2}{6} = \sum_{k=1}^{\infty} \frac{1}{k^2}$.

Glossing over a few issues1, it's a really neat approach, and a similar method using $\cos(x)$ as your function gives a result for the reciprocals of odd squares (namely, they sum to $\frac{\pi^2}{8})$.

I'd be curious to see other proofs of this - preferably at a level I can follow!

* Edited 2017-09-11 to fix something before @realityminus3 spotted it.

Colin

Colin is a Weymouth maths tutor, author of several Maths For Dummies books and A-level maths guides. He started Flying Colours Maths in 2008. He lives with an espresso pot and nothing to prove.

  1. we're playing a bit fast and loose with infinity here []

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