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.

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