Math Ed Blog from Bruce Yoshiwara: Summing 1/n^2

Saturday, November 19, 2011

Here's how Euler evidently first evaluated the sum

$\sum _{n=1}^{\infty} \frac{1}{n^2}=\frac{\pi^2}{6}$

We know the MacLaurin series for sin x, and hence

$\frac{\sin x}{x}=1-\frac{x^2}{3!}+\frac{x^4}{5!}-\cdots$

On the other hand, the zeros of this function are the nonzero integer multiples of pi, and writing the series in a factored form, we obtain

$1-\frac{x^2}{3!}+\frac{x^4}{5!}-\cdots=\left(1-\frac{x}{\pi} \right ) \left(1-\frac{x}{-\pi} \right ) \left(1-\frac{x}{2\pi} \right ) \left ( 1-\frac{1}{-2\pi} \right )\left(1-\frac{x}{3\pi} \right ) \left(1-\frac{x}{-3\pi} \right )$ $=\left(1-\frac{x^2}{\pi^2}\right)\left(1-\frac{x^2}{(2\pi)^2}\right)\left(1-\frac{x^2}{(3\pi)^2}\right)\cdots$