Re: Disturbed by alternating series.. help!
- From: Dave Seaman <dseaman@xxxxxxxxxxxx>
- Date: Tue, 22 Nov 2005 17:20:55 +0000 (UTC)
On 22 Nov 2005 08:55:27 -0800, shevek4@xxxxxxxxx wrote:
> I came across a very scary problem with alternating series, that may
> drive me insane.
> I was under the mistaken impression that the re-ordering of a sum could
> not affect it, i.e. the commutative property of addition. a+b = b+a
> Or even a + -b = -b + a.
That's true. But the sum of an infinite series is not an ordinary sum.
It's actually the limit of the sequence of partial sums.
> However, it was pointed out to me that infinite alternating series do
> not have this property..
> Namely,
> http://mathworld.wolfram.com/RiemannSeriesTheorem.html
> But this seems to throw into question using infinite series as unique
> solutions for differential equations..
> But can we really say that addition is commutative over the reals?
Yes, we can really say that addition is commutative over the reals. But
that statement applies only to addition, not to limits. Addition is a
finite operation.
The answer to the question you intended to ask is that if a series
converges absolutely, then any rearrangement of the series also
converges, and the sum is the same.
The page you referenced is about conditionally convergent series. That
is, a series {a_n} such that sum_n a_n converges in the ordinary sense,
but the series of absolute values sum_n |a_n| diverges. In that case,
the order of the terms can make a difference.
--
Dave Seaman
Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling.
<http://www.commoncouragepress.com/index.cfm?action=book&bookid=228>
.
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