Divergent Series Thinking about divergent series and ways of "summing" them, they seem to fall into two categories (roughly):


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*Series like $\sum_{k=1}^\infty \frac{1}{k}$, which defy all kinds of regularization or summing methods.

*Series which can be summed, in one way or another, like $\sum_{k=1}^\infty k^s$, which for at least $s \geq 1$ can be seen as $\zeta(-s)$.


My question is:


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*Can the groundfields $\mathbb{R}$ or $\mathbb{C}$ be extented in some ways (like the Hyperreals $\mathbb{R}^*$, or the surreals $\mathbb{S}_\mathbb{R}$ or the surcomplex numbers $\mathbb{S}_\mathbb{C}$), such that you can sum all the divergent series (at least the second type from above) in a more rigorous fashion than the bag of tricks that are usually used to explain these results.

*This sounds like a "sort of" completion of the reals with all the summable but divergent series in it. I would like to know if such a beast exist, what its properties are and if the $\sum_{k=1}^\infty k^s=\zeta(-s)$ results can be explained from this.

 A: The answer to your question depends on what properties you want these sums to have, and what properties you want the extended field to have.
If you want to extend the reals in such a way that they satisfy the elementary axioms of algebra (commutativity of addition, etc.), but you want the extended system to include infinite quantities, then you are going to come out with exactly the hyperreal number system that you've already mentioned. (See [*] below for a more formal statement of this.)
But the hyperreals are not necessarily going to have the other properties that you might like. In particular, we would like to be able to say whether or not two divergent series have the same sum. But the standard constructive approach to the hyperreals involves taking equivalence classes of sequences, with the equivalence relation defined by an ultrafilter. Since ultrafilters cannot be explicitly constructed, you can't say in general whether divergent sequences are "approaching" the same hyperreal number. What you can say is whether two convergent sequences are approaching the same real number.
This is a lot of technical stuff about the hyperreals, but it arises from more general considerations about how field axioms work. Basically the thing that stops us from doing what you'd like to do in a completely satisfactory way is that the field axioms only deal with finitely many operations. For example, if associativity of addition applied to infinite sums, then we could rewrite the divergent sum $1-1+1-1+\ldots$ as $1+(-1+1)+(-1+1)+\ldots=1$. Because the field axioms lack this kind of power over infinite processes, you can't really make infinite sums behave like finite ones. The ambiguity in the associativity of the sum $1-1+1+\ldots$ is very much like the ambiguity involved in specifying an ultrafilter.
If you use a system like the surreals, then you have to give up the ability to talk about transcendental functions and do analysis.
[*] The hyperreals are the extension of the reals that you get when you demand the transfer principle for all statements in first-order logic but take the converse of the Archimedean property as an axiom. You also get stuff like transcendental functions, which it's not immediately obvious that you get for free from the transfer principle. The hyperreals are not unique in ZFC, but the different models don't differ in ways that make a difference in this discussion. The hyperreals lack the completeness property of the reals: http://en.wikipedia.org/wiki/Completeness_of_the_real_numbers
A: A related notion is that of a Banach limit. In that case, one does not extend the reals, but one extends the functional that assigns a limit to each sequence. These extensions are not unique, but they exist.
On the other end of the spectrum, you could simply extend the real numbers to all sequences modulo some equivalence relations.
A: I haven't been able to make this idea too formal yet but essential some divergent series yield to power-series while other divergent series seem to explicitly require logarithmic-power-series in order to crack. I think there is a way to put these on a hierarchy where the series $\sum_{n=1}^{\infty} \frac{1}{n}$ lives on a higher level.
So to discover that $1 - 1 + 1 - 1 ... = \frac{1}{2}$  It suffices to look at the series $\frac{1}{1-x}$ and evaluate this for $x = 1$. Similarly you can take derivatives and use tricks to evaluate things like $ 1 - 2  + 3 - 4 ... = \frac{1}{4}$ etc...
Unfortunately as long as you are working with taylor series of rational functions it DOES NOT SEEM that you can discover facts like $ 1  + 2 + 3 + 4...= -\frac{1}{12}$ etc... In order to discover these values one needs to consider functions like $\frac{1}{1-e^x}$  and expand these as a laurent series OR look at the "logarithmic-laurent" expansion of $\frac{1}{1-x}$ around the pole $x=1$. Doing this yields $\frac{1}{1-x} = -\frac{1}{\ln(x)} + \frac{1}{2} - \frac{1}{12}\ln(x) ... $.
Now going after the value $\sum_{n=1}^{\infty} \frac{1}{n}$ is even trickier. Normally the way it is found is by this trick but we might try to use our current tools to get after it. Consider the function
$$ f(x) = -\ln(1-x) = x + \frac{x^2}{2} + \frac{x^3}{3} + ... $$
Clearly $f(1)$ is the value that we want. Since there is a pole there we might want to go after its log series. So we try to extract a laurent series for the expression $g(x) = -\ln(1-e^x)$ centered at $x=0$. No such laurent series exists and instead we end up finding that
$$ g(x) = -\ln(-x) - \frac{x}{2} - \frac{x^2}{24} + \frac{x^4}{2880} ... $$
Okay so this didn't really help at all but we can conclude that $-\ln(0)$ and $\sum_{n=1}^{\infty} \frac{1}{n}$ renormalize to the same magic value whatever that may be. Our wikipedia article using its limit trick says this value is $\gamma$ the Euler Mascheroni Constant.
