  The sum in the title is a rational multiple of pi^n 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: the sum is written as a
definite integral over the unit n-cube, then transformed into the volume of a
polytope Pi_n in R^n whose vertices' coordinates are rational multiples of pi.
We review Calabi's proof, and give two further interpretations. First we define
a simple linear operator T on L^2(0,pi/2), and show that T is self-adjoint and
compact, and that Vol(Pi_n) is the trace of T^n. We find that the spectrum of T
is {1/(4k+1) : k in Z}, with each eigenvalue 1/(4k+1) occurring with
multiplicity 1; thus Vol(Pi_n) is the sum of the n-th powers of these
eigenvalues. We also interpret Vol(Pi_n) combinatorially in terms of the number
of alternating permutations of n+1 letters, and if n is even also in terms of
the number of cyclically alternating permutations of n letters. We thus relate
these numbers with S(n) without the intervention of Bernoulli and Euler numbers
or their generating functions.
