From: kovarik@mcmail.cis.McMaster.CA (Zdislav V. Kovarik)
Subject: Re: infinite series
Date: 30 Jul 1999 22:51:45 -0400
Newsgroups: sci.math
Keywords: axiomatic treatment of infinite series

In article <7ntibp$ge7$1@news-02.meganews.com>, deja <deja@vu.com> wrote:
>
>>the infinite series 1 - 1 + 1 - 1 + 1 - 1 + 1 - ... is somewhat bothersome.
>
>It WAS, (in the 1700 and 1800s)
>:-)
>
>Among the notables that you mentioned, some other people thought (may have
>even been Leibnitz) that it summed to 1/2 !!!!
>
>However, today it is no mystery.

Right on: the confusion starts when we expect things to be where they
aren't. For example, much time and anguish has been spent here on the

"smallest positive real number"

(I advised to the person who claimed to have found it: keep it on one copy
of the real line, and keep its half on another copy, and don't let anyone
see them together. The poor soul didn't get it.:-)=

I could add to it "the smallest even integer n such that (n+1) is also
even".

And "the sum of the alternating 1's and (-1)'s " is in the same bag: just
because you have a description of it, doesn't mean that it must exist (or
that you can wish it into existence).

Yet, there is something called "extended summation operation", and there
is a whole industry about it (Hardy: "Divergent Series"). An extended
summation method is not to be confused with the original, classical
summation method (limit of the sequence of partial sums, if it exists). 

[An extended electrical chord is not to be confused with the original
electrical chord. People concerned about electrical safety can tell you
stories about it.]

One way, of quite limited applicability, is "axiomatic definition" to
extend the notion of the sum: We assume

additivity (sum(a(n) + b(n)) = sum(a(n)) + sum(b(n)) as long as the right
side is defined}

homogeneity (sum(c*a(n)) = c * sum(a(n)) as long as right side is defined,
and c is a number)

shift simplification property (sum(a(n)) = a(1) + sum(a(n+1)))

and nothing more.

Then the axiomatically extended sum of alternating (+1)'s and (-1)'s is
given a name S, and we obtain:

 S = 1 - (1 - 1 + 1 - 1 + ...)

 S = 1 - S

 S + S = 1

 S = 1/2

That's what Leibniz might have seen in it: if we wish the (extended -- a
word he might have omitted) sum to be a number defined consistently with
the linearity and shift property, it must be 1/2. 

To have more fun, let's get the extended sum of 1 + 2 + 4 + 8 + ..., of
the powers of 2 with non-negative integer exponents, in this order.

S = 1 + 2 + 4 + 8 + ... = 1 + 2 * (1 + 2 + 4 + 8 + ...) = 1 + 2 * S

In short, S = 1 + 2 * S , and inevitably this extended (I emphasize) sum
of positive terms equals (-1) (ouch!)

This is a possible price of extending classical concepts beyond their
reasonable "limits".

For other extensions of summation, you can have fun with Cesaro averages,
and with Abel's summation. They are discussed excellently in that book by
Hardy. 

For Axiom of Choice aficionados, there is Banach Limit, covered in
Functional Analysis books, in exercises on Hahn-Banach Theorem.

Cheers, ZVK(Slavek).