Saturday, November 19, 2011

Summing 1/n^2

Here's how Euler evidently first evaluated the sum

We know the MacLaurin series for sin x, and hence

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

Expanding the last (infinite) product of binomials and equating its quadratic coefficient with that of the original MacLaurin series, we obtain

Euler's result follows directly.