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Hugh Alan Grant (umgran22@CC.UManitoba.CA) wrote
:  Hello.  In a calculus textbook I own, it is stated that:
:  1 + 1/4 + 1/9 + 1/16 + ... + 1/(n^2)+ ... = ((pi)^2)/6,
:  1 + 1/8 + 1/27 + 1/64 + ... + 1/(n^3) + ... = 1.2020569031, and
:  1 + 1/16 + 1/81 + 1/256 + ... + 1/(n^4) + ... = ((pi)^4)/90.
: 	Does anyone know how these sums would be calculated?

I can remember being delighted at seeing that done!  But it requires some 
powerful machinery.  The method I saw was an application of "Complex 
Contour Integration".  (Calculus with imaginary numbers.)

The first one is done by integrating the function Cot(pi*z)/z^2 around a 
square in the complex plane and taking the limit as the square expands to 
infinity.

There is a description on pages 127-129 of the book COMPLEX ANALYSIS by 
Joseph Bak and Donald J. Newman published by Springer-Verlag in their 
Undergraduate Texts in Mathematics series.
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