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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: [mg41893] Re: [mg41870] Re: [mg41828] Re: [mg41793] Re: A bug?......In[1]:= Sum[Cos[x], {x, 0, Infinity, Pi}]......Out[1]= 1/2
• From: Bobby Treat <drmajorbob-MathGroup3528 at mailblocks.com>
• Date: Sun, 8 Jun 2003 06:45:54 -0400 (EDT)
• Sender: owner-wri-mathgroup at wolfram.com

What function does that sum represent, then?

Bobby

-----Original Message-----
From: Michael Williams <williams at vt.edu>
To: mathgroup at smc.vnet.net
Subject: [mg41893] [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: [mg41893] 
[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;