End of article 180261 (of 181143) -- what next? [npq] sci.math #179502 (7 + 1580 more) ( )+-[1]
From: doozy@dizzy.mps.ohio-state.edu.--- (Philo D.) \-[1]
[1] Re: Sum of n^-2 (n: 1 to Infinity) = Pi^2 / 6
Date: Fri Feb 14 09:50:26 EST 1997
Lines: 15
In article <01b7ff65$f00f6ae0$05991fc3@antonio>, "Antonio Piazzoni"
wrote:
> I have a problem with the
>
> Sum of 1/n^2 with n from 1 to Infinity.
>
> I know that the sum results Pi^2/6 but I'm not able to show it without the
> Fourier series.
It may be done alternatively with complex analysis (contour integration).
But you are correct that this value is not as elementary as
the sum of a geometric series, for example.
--
*
End of article 179502 (of 181143) -- what next? [npq] sci.math #179523 (6 + 1580 more) ( )+-(1)
Date: Fri Feb 14 09:51:05 EST 1997 \-[1]
From: rjc@maths.ex.ac.uk
[1] Re: Sum of n^-2 (n: 1 to Infinity) = Pi^2 / 6
Lines: 15
In article <01b7ff65$f00f6ae0$05991fc3@antonio>,
"Antonio Piazzoni" wrote:
>
> I have a problem with the
>
> Sum of 1/n^2 with n from 1 to Infinity.
>
> I know that the sum results Pi^2/6 but I'm not able to show it without the
> Fourier series.
>
If you can handle dvi see
http://www.maths.ex.ac.uk/~rjc/etc/zeta2.dvi .
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End of article 179523 (of 181143) -- what next? [npq] sci.math #179506 (5 + 1580 more) ( )+-[1]
From: asari@math.uiuc.edu (ASARI Hirotsugu) |-[1]
[1] Re: Sum of n^-2 with n from 1 to infinity = Pi^2/6 . \-[1]--[1]+-[1]
+ .. \-[1]
Date: Fri Feb 14 09:19:05 EST 1997
I've seen it done with the identity.
We have $\sin \pi z=\pi z \prod_{n=1}^\infty (1-z^2/n^2)$ and the
Maclaurin series for $\sin \pi z$, $\pi z - {(\pi z)^3\over 3!}+...$.
Expand the former and compare the $z^3$ term.
--
ASARI Hirotsugu // http://www.math.uiuc.edu/~asari/
finger://math.uiuc.edu/asari // ph://ns.uiuc.edu/asari
In the Zen of programming, you are what
you program, and your code is you. --Jim Trudeau
End of article 179506 (of 181143) -- what next? [npq] sci.math #179628 (4 + 1580 more) ( )+-(1)
From: ptwahl@aol.com (PTWahl) |-[1]
[1] Re: Sum of n^-2 with n from 1 to infinity = Pi^2/6 . \-[1]--[1]+-[1]
Date: Sat Feb 15 04:34:50 EST 1997
There are several proofs of this fact.
A short one (via complex variables) uses the residue theorem, and is shown
in many texts. It is available, for example, in the Schaum's outline
title, "Complex Variables", by Murray R. Spiegel. Solved Problem #28 of
Chapter 7 outlines the proof, and hints at another.
Nor need we travel beyond real variables: Tom Apostol published a proof
using double integrals and coordinate system changes. It's comprehensible
at the Calculus III level; quite clever, though (in my opinion) not at all
straightforward. One feature is the long bibliography of other proofs. I
could dig up the reference if you are interested.
Patrick T. Wahl
( no institutional affiliation )
End of article 179628 (of 181143) -- what next? [npq] sci.math #179633 (3 + 1580 more) ( )+-(1)
From: mckay@cs.concordia.ca (MCKAY john) |-(1)
[1] Re: Sum of n^-2 with n from 1 to infinity = Pi^2/6 . \-[1]--[1]+-[1]
Date: Sat Feb 15 05:09:30 EST 1997
The beautiful proof below is Euler's - today it can be justified! The
evaluation of zeta(2) was regarded as a tough call before Euler's proof:
x*prod(1-x^2/(n*pi)^2) = sin(x) [from zeros] = x - x^3/3! + x^5/5! +-...
Comparing coefficient of x^3, we find:
-sum(1/(n*pi)^2) = -1/3!
Try other coefficients!
JM
--
Cogito ergo sum aut miror ergo sim?
End of article 179633 (of 181143) -- what next? [npq] sci.math #179689 (2 + 1580 more) ( )+-(1)
From: bruck@pacificnet.net (Ronald Bruck) |-(1)
[1] Re: Sum of n^-2 with n from 1 to infinity = Pi^2/6 . \-(1)--[1]+-[1]
Date: Sat Feb 15 16:56:46 EST 1997
Yes, it's a beautiful way to do it. But the development of sin(z) into the
infinite product isn't straightforward; you can multiply it by exp(f(z))
for any entire function f, and have another analytic function with the same
zeros.
It takes a little work to show what the RIGHT multiplier is.
Has anyone read Euler's original? Did he take care of this detail, or did
he just make the guess?
--Ron Bruck
Now 100% ISDN from this address
End of article 179689 (of 181143) -- what next? [npq] sci.math #179734 (1 + 1580 more) ( )+-(1)
From: ikastan@alumnae.caltech.edu (Ilias Kastanas) |-(1)
[1] Re: Sum of n^-2 with n from 1 to infinity = Pi^2/6 . \-(1)--(1)+-[1]
Date: Sun Feb 16 03:39:11 EST 1997
Well, I did read "Intro. in anal. infinitorum" some time ago, and
I think I saw this there... without what we would call today a rigorous
all, written in 1748.
An elementary approach to zeta(2) was given by Eisenstein. Consider
f(x) = 1/x + Sum[over n > 0] 2x/(x^2 - n^2). Calculate f(x)^2, using the
fact that Sum[ m>0, m != n] 1/(m^2 - n^2) = 3/4n^2, and thus show
f'(x) + f(x)^2 = 6 Sum[n >= 1] 1/n^2. Prove that f(x) = pi cot(pi x), and
zeta(2) = pi^2 /6 follows... without Fourier series or complex analysis.
Ilias
End of article 179734 (of 181143) -- what next? [npq] sci.math #180344 (0 + 1580 more) ( )+-(1)
From: |-(1)
[1] Re: Sum of n^-2 with n from 1 to infinity = Pi^2/6 . \-(1)--(1)+-(1)
Date: Wed Feb 19 10:51:25 EST 1997
The history of Euler's proof and his attempts to make it rigorous are
Plausible Reasoning_ vol 1.
Regards,
Dipankar
--
Dipankar Gupta _ @ HP . COM
Hewlett-Packard Laboratories, Bristol UK <>
PGP fingerprint: 64 77 3C 86 14 1F 1A 9F BA 27 83 36 7D D4 BE 71
End of article 180344 (of 181143) -- what next? [npq]