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From: tim@ism780c.UUCP (Tim Smith)
Newsgroups: net.math,net.physics
Subject: Re: value of an integral
Message-ID: <796@ism780c.UUCP>
Date: Thu, 27-Feb-86 15:31:57 EST
Article-I.D.: ism780c.796
Posted: Thu Feb 27 15:31:57 1986
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References: <823@drux2.UUCP> <896@yale.ARPA>
Reply-To: tim@ism780c.UUCP (Tim Smith)
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Xref: watmath net.math:2901 net.physics:3891

In article <896@yale.ARPA> makdisi@yale-cheops.UUCP (Kamal Khuri-Makdisi) writes:
>
>The only proof I know that the sum of 1/n^4, n = 1 to infinity, is pi^2/90
>involves Fourier series -- anyone know a more elementary way of proving this?
>
It depends on what you consider elementary.  In "An Introduction 
to Analytic Number Theory", by Tom Apostol, he evaluates Zeta(2n) 
for positive integers n.  This is done in chapter 12. Your series
is Zeta(4).  He has to use contour integration.  I think it is more
elementary than Fourier series.

-- 
Tim Smith       sdcrdcf!ism780c!tim || ima!ism780!tim || ihnp4!cithep!tim