From: David Bernier <bernier@my-deja.com>
Subject: Re: sum(exp(-kČ/4),k=0..infinity)=sqrt(Pi)+1/2 is true?
Date: Wed, 02 Feb 2000 12:47:02 GMT
Newsgroups: sci.math
Summary: [missing]

In article <8791t1$56c$1@nnrp1.deja.com>,
  Robin Chapman <rjc@maths.ex.ac.uk> wrote:
> In article <878ts3$17h$1@front5.grolier.fr>,
>   "Panh" <panh@club-internet.fr> wrote:
> > How prove
>
> > sum(exp(-kČ/4),k=0..infinity)=sqrt(Pi)+1/2 is true?
>
> I isn't. The difference between the sides is about 2.5 x 10^{-15}.

Using the ISC calculator [1], I approximated:
(a) 1/2 + sum(exp(-kČ/4),k=1..infinity)  and
(b)  sqrt(pi).

(a) evaluates to:  1.7724538509055160526696597756308    and
(b) evaluates to:  1.7724538509055160272981674833411

while (a)-(b) evaluates to:  0.253714922922897e-16

intriguing...

David Bernier
--
[1]  http://www.cecm.sfu.ca/projects/ISC/ISCmain.html



Sent via Deja.com http://www.deja.com/
Before you buy.
==============================================================================

From: Robin Chapman <rjc@maths.ex.ac.uk>
Subject: Re: sum(exp(-kČ/4),k=0..infinity)=sqrt(Pi)+1/2 is true?
Date: Wed, 02 Feb 2000 13:55:22 GMT
Newsgroups: sci.math

[Essentially all of previous message quoted --djr]

See Problem 81.F in the Mathematical Gazette (July 1997).

--
Robin Chapman, http://www.maths.ex.ac.uk/~rjc/rjc.html
 "`The twenty-first century didn't begin until a minute
  past midnight January first 2001.'"
   John Brunner, _Stand on Zanzibar_ (1968)


Sent via Deja.com http://www.deja.com/
Before you buy.
==============================================================================

From: "G. A. Edgar" <edgar@math.ohio-state.edu.nospam>
Subject: Re: sum(exp(-kČ/4),k=0..infinity)=sqrt(Pi)+1/2 is true?
Date: Wed, 02 Feb 2000 09:16:04 -0500
Newsgroups: sci.math

[Essentially all of prior message quoted --djr]

I agree they are not equal.  The sum is essentially a theta
function...  Maple's definition is


                                /infinity                       \
                                | -----      2                  |
                                |  \       (n )                 |
     JacobiTheta3(z, q) = 1 + 2 |   )     q     cos((2 n + 1) z)|
                                |  /                            |
                                | -----                         |
                                \ n = 1                         /

so the sum here is JacobiTheta3(0,exp(-1/4))/2 =
 1.7724538509055160526696597756309300872004041978324...,
close but not equal to sqrt(pi) = 
 1.7724538509055160272981674833411451827975494561224...

-- 
Gerald A. Edgar              edgar@math.ohio-state.edu