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
**********
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