Services & ResourcesWolfram ForumsMathGroup Archive

[Date Index] [Thread Index] [Author Index]

Re: 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: [mg41895] Re: [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: Sun, 8 Jun 2003 06:45:56 -0400 (EDT)
• Sender: owner-wri-mathgroup at wolfram.com

1/(1-z) of course.  The point is its power series representation (at 
z=0) diverges for ALL z on the unit circle (in the complex plane).  If 
we examine the series, using Cesaro sums, we get values for all z on 
the unit circle except z=1 (where the real trouble is!), AND those 
values agree with 1/(1-z).  The values "formally" obtained remain 
faithful to the given function (defined on the whole plane).

Michael

On Saturday, June 7, 2003, at 08:54 PM, Bobby Treat wrote:

> What function does that sum represent, then?
>
> Bobby
>
> -----Original Message-----
> From: Michael Williams <williams at vt.edu>
To: mathgroup at smc.vnet.net
> To: mathgroup at smc.vnet.net
> Sent: Sat, 7 Jun 2003 11:44:55 -0400 (EDT)
> Subject: [mg41895] [mg41870] Re: [mg41828] Re: [mg41793] Re: A bug?......In[1]:= 
> Sum[Cos[x], {x, 0, Infinity, Pi}]......Out[1]= 1/2
>
> 1/(1-z)=Sum[z^n,{n,0,Infinity}] |z|&lt;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-&gt;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 (&quot;formal sum&quot;) 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: &gt; Sum[Cos[x],{x,0,Infinity,Pi}] doesn't converge in 
> any sense that's &gt; useful to most of us, and I'm curious what kind 
> of analysis would &gt; benefit from assuming that it does converge 
> somehow. &gt; &gt; Dana's computations show how easy it is to formally 
> &quot;prove&quot; that it &gt; converges, however, if we misapply a 
> method that often works. &gt; &gt; Bobby &gt; &gt; -----Original 
> Message----- &gt; From: Dana DeLouis &lt;delouis at bellsouth.net&gt; To: 
To: mathgroup at smc.vnet.net
> mathgroup at smc.vnet.net &gt; To: mathgroup at smc.vnet.net &gt; Subject: [mg41895] 
> [mg41870] [mg41828] [mg41793] Re: A bug?......In[1]:= Sum[Cos[x], {x, 
> &gt; 0, Infinity, &gt; Pi}]......Out[1]= 1/2 &gt; &gt; Hello. I am not 
> an expert, but I came across a chapter recently in my &gt; studies of 
> Fourier Analysis. Basically, your series sums the following &gt; 
> terms. (the first 10 terms...) Table[Cos[x], {x, 0, 10*Pi, Pi}] {1, 
> -1, &gt; 1, -1, 1, -1, 1, -1, 1, -1, 1} You are summing a series of 
> alternating &gt; +1 and -1's. Your series can also be written like 
> this... Plus @@ &gt; Table[(-1)^j*r^j, {j, 0, 10}] 1 - r + r^2 - r^3 + 
> r^4 - r^5 + r^6 - r^7 &gt; + r^8 - r^9 + r^10 With r equal to 1 For 
> example, if r is 1, then the &gt; first 10 terms are... 
> Table[(-1)^j*r^j, {j, 0, 10}] /. r -&amp;gt; 1 {1, &gt; -1, 1, -1, 1, 
> -1, 1, -1, 1, -1, 1} If you sum this as j goes to &gt; infinity, you 
> get the following. Sum[(-1)^j*r^j, {j, 0, Infinity}] 1/(1 &gt; + r) 
> Apparently, this is correct and has something to do with Abel's &gt; 
> method. I still do not understand this topic too well yet though. &gt; 
> Anyway, if you set r = 1, then 1/(1+r) reduces to 1/2. Although it 
> &gt; doesn't look like it, I believe Mathematica is correct -- Dana 
> DeLouis &gt; Windows XP Mathematica $VersionNumber -&amp;gt; 4.2 
> delouis at bellsouth.net = &gt; = = = = = = = = = = = = = = = = 
> &amp;quot;Mark&amp;quot; &gt; &amp;lt;nanoburst at yahoo.com&amp;gt; 
> wrote in message &gt; news:bb1ua4$9do$1 at smc.vnet.net... &amp;gt; I 
> think that the sum does not &gt; converge. Does &amp;gt; the following 
> (from Mathematica for Students, &amp;gt; &gt; v. 4.0.1) reveal a bug? 
> If so, do you have &amp;gt; any insight into this &gt; bug? &amp;gt; 
> &amp;gt; &amp;gt; In[1]:= Sum[Cos[x], {x, 0, Infinity, Pi}] &amp;gt; 
> &amp;gt; &gt; Out[1]= 1/2 &amp;gt; &amp;gt; &amp;gt; &amp;gt; &amp;gt; 
> &amp;gt; ********** &amp;gt; 1366294709 &gt; &amp;gt; &gt;