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

• To: mathgroup at smc.vnet.net
• Subject: [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: Fri, 6 Jun 2003 09:51:00 -0400 (EDT)
• Sender: owner-wri-mathgroup at wolfram.com

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
Subject: [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;