On the Sums ∞ K=−∞ (4k + 1)


The sum in the title is a rational multiple of π for all integers n = 2, 3, 4, . . . for which the sum converges absolutely. This is equivalent to a celebrated theorem of Euler. Of the many proofs that have appeared since Euler, a simple one was discovered only recently by Calabi [4]: the sum is written as a definite integral over the unit n-cube, then… (More)


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