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This question arose after I saw a youtube-vid where Grandi's series was discussed.
It seems that the sum of the series will be 0 for an even, and 1 for an odd number of terms, where a term is defined as (-1)n, n indicating the n'th term.
It seems that even when s is derived that the above tacit assumptions of even/odd is used.
Is it valid to make these assumptions?

(My lay-opinion is that these assumptions are wrong, and therefor also the derived value of s (= 1/2), and that the series has no sum as stated in the wikipedia entry)

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You can either consider it as undefined to even ask about whether an infinite set is even or odd, or you can say that it is both even and odd. If you define "odd" as "not even," the it could also be considered even, and hence not odd. It all depends on how you define things. –  Thomas Andrews Jun 27 '13 at 16:55
    
@ThomasAndrews: the sole purpose of definitions are to support arguments for some model followed by the usual falsifiability tests; in Grandi's there are no such stated definitions and the model that is arrived at does not explain nor predict, which makes the whole exercise not much better than nonsense - which was my reaction to it and that prompted the question in case I'm missing something fundamental. –  slashmais Jun 28 '13 at 7:17
    
Umm, no, that isn't even remotely the purpose of definitions. The most common purpose is so that you don't have to repeat common phrases over and over again. For example, if "$n$ is even" means $\exists m: n=m+m$, then most of the time, I'd rather write "$n$ is even" than the long form. –  Thomas Andrews Jun 28 '13 at 11:10
    
@ThomasAndrews: to avoid confusion, to provide common understanding in communication (oral or written), to reduce verbiage, ... always used within some argument around some concept (=model) - there is no other purpose or need for definitions. –  slashmais Jun 28 '13 at 12:16

1 Answer 1

up vote 2 down vote accepted

You are exactly right. An "infinite number of terms" is neither even nor odd. With Grandi's series, you can only discuss the values of partial sums of the series - because the series itself does not converge, and therefore does not have any value.

A series is defined as the limit of its sequence of partial sums, if that limit exists: $$ \sum_{n=0}^{\infty}a_n=\lim_{N\rightarrow\infty}\sum_{n=0}^{N}a_n $$ For Grandi's series, the sequence of partial sums alternates between 0 and 1, and does not converge; hence the series itself is undefined.

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the limit is undefined - not the series. –  Tomas Jun 27 '13 at 16:57
    
I disagree. Clearly, the terms of the series are defined - but the series itself cannot be assigned a numerical value in any reasonable way. I'd call that 'undefined'. –  Nicholas R. Peterson Jun 27 '13 at 16:59
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I agree with Tomas: as far as we have a sequenece, any sequence, its series exists. It is its sum, not the series, that is defined as the limit of the sequence of the series' partial sums in case this limit exists finitely, –  DonAntonio Jun 27 '13 at 17:18
    
I can respect that. At the end of the day, this is a difference which is really only terminological, not functional... and stems from how you choose to define a series. I define it as the limit of its sequence of partial sums, unless I'm in the land of formal series... hence my preference. But, I also completely understand where you're coming from! –  Nicholas R. Peterson Jun 27 '13 at 18:08
    
Sorry, @nrpeterson, I was not familiar with your terminology and assumed, it was typo. –  Tomas Jun 27 '13 at 18:37

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