Re: A bug?......In[1]:= Sum[Cos[x], {x, 0, Infinity, Pi}]......Out[1]= 1/2
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- Subject: [mg41793] Re: A bug?......In[1]:= Sum[Cos[x], {x, 0, Infinity, Pi}]......Out[1]= 1/2
- From: "Dana DeLouis" <delouis at bellsouth.net>
- Date: Thu, 5 Jun 2003 07:31:35 -0400 (EDT)
- Sender: owner-wri-mathgroup at wolfram.com
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
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"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
>