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|<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: To: mathgroup at smc.vnet.net > mathgroup at smc.vnet.net > To: mathgroup at smc.vnet.net > Subject: [mg41895] > [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; >