Re: Leibniz Definition Of Pi Not In 5.0.0?
- To: mathgroup at smc.vnet.net
- Subject: [mg43290] Re: Leibniz Definition Of Pi Not In 5.0.0?
- From: Andrzej Kozlowski <andrzej at platon.c.u-tokyo.ac.jp>
- Date: Sat, 23 Aug 2003 08:09:24 -0400 (EDT)
- Sender: owner-wri-mathgroup at wolfram.com
The explanation is that Mathematica evaluates If[EvenQ[n], -(1/(2*n -
1)), 1/(2*n - 1)] where n is a symbol, not an integer. EvenQ[n] is
False for a symbol n, so you get 1/(2*n - 1) as the output. Your
expression becomes simply:
Pi/4 === Sum[1/(-1 + 2*n), {n, 1, Infinity}]
and of course the right hand side does not converge, so you get the
result.
You can see it all for yourself using Trace.
Andrzej Kozlowski
Yokohama, Japan
http://www.mimuw.edu.pl/~akoz/
http://platon.c.u-tokyo.ac.jp/andrzej/
On Friday, August 22, 2003, at 12:12 AM, H. Burke Jensen wrote:
> Thank you All for the better code!
>
> My question still stands though regarding why using this method the
> Sum is
> incorrectly identified. I'm trying to figure out the internal thinking
> Mathematica does to discover where the difference drives an incorrect
> identification. Any ideas?
>
> Thank you again,
> -H. Burke Jensen
> hbj at ColoradoKidd.com
> The Colorado Kidd®
> www.ColoradoKidd.com
>
> "H. Burke Jensen" <hbj at ColoradoKidd.com> wrote in message
> news:bht3v3$n4n$1 at smc.vnet.net...
> $Version: 5.0 for Microsoft Windows (June 10, 2003)
> $MachineType: PC
> $OperatingSystem: WindowsNT
>
> Hello MathGroup,
>
> Does Mathematica 5.0.0 not recognize the Leibniz definition of Pi
> [Ref.1]?
> This was recognized in Mathematica 3.0.1 and reported to WRI.
>
> In[1]:=
> \!\(\[Pi]\/4 === \[Sum]\+\(n = 1\)\%\[Infinity] If[
> EvenQ[n] \[Equal] True, \(-\(1\/\(2 n - 1\)\)\), 1\/\(2 n -
> 1\)]\)
>
> Sum::div: Sum does not converge.
>
> Sum::div: Sum does not converge.
>
> Out[1]=
> False
>
> References:
> [1] Martin, George E., The Foundations of Geometry and the
> Non-Euclidean
> Plane, Springer, 1975, p. 157-158.
>
> Thank you,
> -H. Burke Jensen
> hbj at ColoradoKidd.com
> The Colorado Kidd®
> www.ColoradoKidd.com