Working with the set of real numbers (was: Finding size of Variable)

[Oscar Benjamin]

On 5 March 2014 07:52, Steven D'Aprano <steve at pearwood.info> wrote:
> On Tue, 04 Mar 2014 23:25:37 -0500, Roy Smith wrote:
>
>> I stopped paying attention to mathematicians when they tried to convince
>> me that the sum of all natural numbers is -1/12.
>
> I'm pretty sure they did not. Possibly a physicist may have tried to tell
> you that, but most mathematicians consider physicists to be lousy
> mathematicians, and the mere fact that they're results seem to actually
> work in practice is an embarrassment for the entire universe. A
> mathematician would probably have said that the sum of all natural
> numbers is divergent and therefore there is no finite answer.

Why the dig at physicists? I think most physicists would be able to
tell you that the sum of all natural numbers is not -1/12. In fact
most people with very little background in mathematics can tell you
that.

The argument that the sum of all natural numbers comes to -1/12 is
just some kind of hoax. I don't think *anyone* seriously believes it.

> Well, that is, apart from mathematicians like Euler and Ramanujan. When
> people like them tell you something, you better pay attention.

Really? Euler didn't even know about absolutely convergent series (the
point in question) and would quite happily combine infinite series to
obtain a formula.

<snip>
> Normally mathematicians will tell you that divergent series don't have a
> total. That's because often the total you get can vary depending on how
> you add them up. The classic example is summing the infinite series:
>
> 1 - 1 + 1 - 1 + 1 - ...

There is a distinction between absolute convergence and convergence.
Rearranging the order of the terms in the above infinite sum is
invalid because the series is not absolutely convergent. For this
particular series there is no sense in which its sum converges on an
answer but there are other series that cannot be rearranged while
still being convergent:
http://en.wikipedia.org/wiki/Harmonic_series_(mathematics)#Alternating_harmonic_series

Personally I think it's reasonable to just say that the sum of the
natural numbers is infinite rather than messing around with terms like
undefined, divergent, or existence. There is a clear difference
between a series (or any limit) that fails to converge  asymptotically
and another that just goes to +-infinity. The difference is usually
also relevant to any practical application of this kind of maths.


Oscar