How are infinite sums in nonstandard analysis defined?

Question

Since, in nonstandard one can have infinitely large numbers, I was wondering if I can assign divergent sums to them. However infinite sums are defined by taking the limit to infinity of a partial sum of the sequence. But would that mean all divergent sums that go to $\infty$ are the same infinite number, or is this some kind of a different infinity? Can I even use the limit here?

Answer 1

We can apply the transfer principle to the fact that finite sums of arbitrary sequences exist. More precisely,

For any internal sequence $x_n$ of hyperreal numbers, defined over hyperintegers $n = a, a+1, \ldots, b$, the sum $$ \sum_{n=a}^b x_n $$ is well-defined.

Now, the thing to pay attention to is that this sum only works for sequences that have a beginning and an end. For example, given any infinite hyperinteger $H$, we have

$$ \sum_{n=0}^H n = \frac{H(H+1)}{2} $$

but we cannot use this to make sense of a sum like

$$ \sum_{n \in \mathbb{N}} n = {???}$$

Note, incidentally, that the transfer principle can also be applied to infinite summation; we can define what it means for there to be a sum $\sum_{n=a}^{\infty} x_n$ for internal sequences $x_n$. But note this sum is over all hyperintegers greater than or equal to $a$. (In particular, $\infty$ is bigger than every hyperreal number)

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Another thing you can do is define a brand new form of summation, which we might call the "ultrasum", by taking the "ultralimit" of the partial sums.

In particular, given the specific model of the hyperreals in terms of ultrapowers then given any sequence of ordinary reals $x_n$ for $n=0,1,\ldots$, we can define its ultrasum to be the hyperreal

$$ \left( \sum_{n=0}^0 x_n, \sum_{n=0}^1 x_n, \sum_{n=0}^2 x_n, \ldots \right) $$

Incidentally, if $H$ is the hyperreal corresponding to $(0, 1, 2, \ldots)$ and we use the sequence $x_n = n$, I think this ultrasum is the same value as the sum $\sum_{n=0}^H n$ given above.

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For a brief sketch of how the definition above works, the point is that in real analysis, we can define a set $S$ of all finite sequences, and summation is a specific function from $S \to \mathbb{R}$.

The hyperfinite sum defined above is precisely the transfer of this function, which is an internal function ${}^\star S \to {}^\star \mathbb{R}$.

Answer 2

You may want to consult the elementary treatment found in Keisler's textbook Elementary Calculus.

Two of the main principles at work here are the Extension Principle and the Transfer Principle.

The Extension Principle asserts that any real function has a natural extension; for example, a sequence $u=(u_n)$ of real numbers can be viewed as a function $u:\mathbb N \to \mathbb R$, and therefore has a natural extension $u: {}^\ast \mathbb N \to {}^\ast \mathbb R$.

Thus means that the extended sequence now has terms labeled by all hypernatural numbers, say $H$, as well, so that $u_H$ is well defined. For example if the sequence tends to infinity then the term $u_H$ will be infinitesimal for all infinite $H$. This is shown using the Transfer Principle.

If a sequence $(u_n)$ tends to infinity then $u_H$ will be an infinite hyperreal number for all infinite values $H$ of the index.

The limit of a sequence is defined as the standard part of $u_H$ for infinite $H$. If $u_H$ happens to be infinite then the standard part is not defined and the limit does not exist.

This also provides an explanation of "infinite sums" (more precisely, hyperfinite sums) as follows. Denote by $S_n$ the partial sum of the series defined by the sequence $(u_n)$, so that $S_1=u_1$, $S_2=u_1+u+2, S_3=u_1+u_2+u_3$, etc. Then the sequence of partial sums $(S_n)$ has a natural extension by the Extension Principle. Thus $S_H$ is now defined for an infinite hypernatural $H$. This is the kind of "infinite sum" you have in Robinson's framework. If the series converges then its sum is the standard part of $S_H$.