Kjetil et al,
Unlike finite sums, infinite sums are not commutative. To have 
commutativity, one must have absolute summability, that is, the sum of the 
absolute values of the terms must be finite.  If one has absolute 
summability, the infinite sum exists and is unique. This sum is not 
absolutely summable and thus undefined.   If one does not require 
commutativity, then the order of the summation must be specified.  The 
order is often implicitly assumed to be the order of the integers. The sum 
of the negative integers is negative infinity, the sum of the positive 
integers  is infinity, and the sum of these two sums is undefined. 
However, Riemann's rearrangement theorem shows that the terms can be 
re-ordered to yield any sum whatsoever.  In particular, if one creates 
pairs of terms consisting of  a positive integer and its negative, then 
the infinite sum is zero.   So the unique sum is undefined; otherwise the 
sum depends on the order of addition.
Joe



David Winsemius <dwinsemius@comcast.net> 
Sent by: r-help-bounces@r-project.org
02/15/2011 09:17 AM

To
Kjetil Halvorsen <kjetilbrinchmannhalvorsen@gmail.com>
cc
r-help@r-project.org, Maithula Chandrashekhar 
<m.chandrashekhar1977@gmail.com>
Subject
Re: [R] A Math question







On Feb 14, 2011, at 7:33 PM, Kjetil Halvorsen wrote:

> or even better:
>
> http://mathoverflow.net/

I beg to differ. That is designated in its FAQ as expecting "research 
level questions", while the forum I offered is labeled as "Welcome to 
Q&A for people studying math at any level and professionals in related 
fields". I don't think the proffered question could be considered 
"research level".

>
> On Sun, Feb 13, 2011 at 8:02 PM, David Winsemius <dwinsemius@comcast.net 

> > wrote:
>>
>> On Feb 13, 2011, at 4:47 PM, Maithula Chandrashekhar wrote:
>>
>>> Dear all, I admit this is not anything to do R and even with
>>> Statistics perhaps. Strictly speaking this is a math related 
>>> question.
>>> However I have some reasonable feeling that experts here would 
>>> come up
>>> with some elegant suggestion to my question.
>>>
>>> Here my question is: What is sum of all Integers? I somewhere heard
>>> that it is Zero as positive and negative integers will just cancel
>>> each other out. However want to know is it correct?
>>
>> There are more appropriate places to pose such questions:
>> http://math.stackexchange.com/
>>

David Winsemius, MD
West Hartford, CT

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