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: [mg41898] 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 06:46:06 -0400 (EDT)
- Sender: owner-wri-mathgroup at wolfram.com
How about 1/(1+x) at x=1 or x/(2 + x) at 2? In both cases (and infinitely many others), the function is well-behaved there but the series (the very SAME series) is divergent. I gather that Cesaro convergence is indicative of that, and that's good to know. (I guess.) However, as I suspected, the series doesn't represent any function at all. At best, it represents a huge family of functions. Bobby -----Original Message----- From: Michael Williams <williams at vt.edu> To: mathgroup at smc.vnet.net Subject: [mg41898] Re: [mg41870] Re: [mg41828] Re: [mg41793] Re: A bug?......In[1]:= Sum[Cos[x], {x, 0, Infinity, Pi}]......Out[1]= 1/2 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 Subject: [mg41898] [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 To: mathgroup at smc.vnet.net DeLouis delouis at bellsouth.net To: mathgroup at smc.vnet.net To: mathgroup at smc.vnet.net Subject: [mg41898] [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 - 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 - 4.2 delouis at bellsouth.net = = = = = = = = = = = = = = = = = Mark nanoburst at yahoo.com wrote in message news:bb1ua4$9do$1 at smc.vnet.net... I think that the sum does not converge. Does the following (from Mathematica for Students, v. 4.0.1) reveal a bug? If so, do you have any insight into this bug? In[1]:= Sum[Cos[x], {x, 0, Infinity, Pi}] Out[1]= 1/2 ********** 1366294709