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Re: Re: Re: A bug?......In[1]:= Sum[Cos[x], {x, 0, Infinity, Pi}]......Out[1]= 1/2

• To: mathgroup at smc.vnet.net
• Subject: [mg41859] Re: [mg41828] Re: [mg41793] Re: A bug?......In[1]:= Sum[Cos[x], {x, 0, Infinity, Pi}]......Out[1]= 1/2
• From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
• Date: Sat, 7 Jun 2003 00:08:48 -0400 (EDT)
• Sender: owner-wri-mathgroup at wolfram.com

There is a whole huge area of mathematics that deals with divergent 
series of this kind.
The most famous classic texts are by G.H. Hardy, "Divergent Series"  
and by Antoni Zygmund  "Trigonometric Series".


Andrzej Kozlowski
Yokohama, Japan
http://www.mimuw.edu.pl/~akoz/
http://platon.c.u-tokyo.ac.jp/andrzej/



On Friday, June 6, 2003, at 10:51  pm, Bobby Treat wrote:

> Sum[Cos[x],{x,0,Infinity,Pi}] doesn't converge in any sense that's
> useful to most of us, and I'm curious what kind of analysis would
> benefit from assuming that it does converge somehow.
>
> Dana's computations show how easy it is to formally "prove" that it
> converges, however, if we misapply a method that often works.
>
> Bobby
>
> -----Original Message-----
> From: Dana DeLouis <delouis at bellsouth.net>
To: mathgroup at smc.vnet.net
> To: mathgroup at smc.vnet.net
> Subject: [mg41859] [mg41828] [mg41793] Re: A bug?......In[1]:= Sum[Cos[x], {x, 
> 0, Infinity,
> Pi}]......Out[1]= 1/2
>
> Hello. I am not an expert, but I came across a chapter recently in my
> studies of Fourier Analysis. Basically, your series sums the following
> terms. (the first 10 terms...) Table[Cos[x], {x, 0, 10*Pi, Pi}] {1, -1,
> 1, -1, 1, -1, 1, -1, 1, -1, 1} You are summing a series of alternating
> +1 and -1's. Your series can also be written like this... Plus @@
> Table[(-1)^j*r^j, {j, 0, 10}] 1 - r + r^2 - r^3 + r^4 - r^5 + r^6 - r^7
> + r^8 - r^9 + r^10 With r equal to 1 For example, if r is 1, then the
> first 10 terms are... Table[(-1)^j*r^j, {j, 0, 10}] /. r -&gt; 1 {1,
> -1, 1, -1, 1, -1, 1, -1, 1, -1, 1} If you sum this as j goes to
> infinity, you get the following. Sum[(-1)^j*r^j, {j, 0, Infinity}] 1/(1
> + r) Apparently, this is correct and has something to do with Abel's
> method. I still do not understand this topic too well yet though.
> Anyway, if you set r = 1, then 1/(1+r) reduces to 1/2. Although it
> doesn't look like it, I believe Mathematica is correct -- Dana DeLouis
> Windows XP Mathematica $VersionNumber -&gt; 4.2 delouis at bellsouth.net =
> = = = = = = = = = = = = = = = = &quot;Mark&quot;
> &lt;nanoburst at yahoo.com&gt; wrote in message
> news:bb1ua4$9do$1 at smc.vnet.net... &gt; I think that the sum does not
> converge. Does &gt; the following (from Mathematica for Students, &gt;
> v. 4.0.1) reveal a bug? If so, do you have &gt; any insight into this
> bug? &gt; &gt; &gt; In[1]:= Sum[Cos[x], {x, 0, Infinity, Pi}] &gt; &gt;
> Out[1]= 1/2 &gt; &gt; &gt; &gt; &gt; &gt; ********** &gt; 1366294709
> &gt;
>
>
>