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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: [mg41870] Re: [mg41828] Re: [mg41793] Re: A bug?......In[1]:= Sum[Cos[x], {x, 0, Infinity, Pi}]......Out[1]= 1/2
• From: Michael Williams <williams at vt.edu>
• Date: Sat, 7 Jun 2003 11:44:55 -0400 (EDT)
• Sender: owner-wri-mathgroup at wolfram.com

1/(1-z)=Sum[z^n,{n,0,Infinity}]   |z|<1

lhs at z=-1     = 1/2

rhs at z=-1    = 1-1+1-1+1-...

The Cesaro sum (e.g.) of a series, u1+u2+u3+...  with partial sums, 
s1,s2,s3, is defined to be the limit as n->Infinity of 
(s1+s2+s3+...+sn)/n .   When a series converges, the Cesaro value is 
the same as the series sum.  It is easy to see that the Cesaro sum of 
the above series is 1/2 and is the correct value for the function that 
the series represents.  Indeed, this is true for all |z|=1, z!=1.

The generalized sum ("formal sum") provides useful (i.e. correct) 
information about the function  the series represents, even when the 
series does not converge in the traditional sense.

Michael Williams
Blacksburg,Va,USA

On Friday, June 6, 2003, at 09:51 AM, 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: [mg41870] [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;
>