Re: Disturbed by alternating series.. help!


John Coleman wrote:
> shevek4@xxxxxxxxx wrote:
> > 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.
> > >
> > > 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?
> >
> > addendum:
> >
> > We've all been told that
> >
> > 1/(1+r) = \sum_i (-r)^i
> >
> > but now Riemann says that the RHS could equal anything at all!
> >
> > even orthogonal polynomials are suspect now.. i.e. infinite Fourier
> > series with coefficients that could be negative..
>
> you are forgetting about absolute convergence. conditionally convergent
> series can be somewhat pathological, but power series are absolutely
> convergent (hence don't change their values upon rearrangement) inside
> their radius of convergence.

But some power series are alternating! I gave an example of the power
series for the function 1/(1+r).

I'm worried that there's a problem with f(x) = \sum_i [ d^i f(x_0)/dx^i
(x-x0)^i / i!]

The derivitives in the Taylor series above might be alternating.. in
which case the sum is meaningless??

> (by the way - commutativity is a property
> of finite sums, one can't expect infinite sums to behave the same way
> in all cases)

OK, I guess I can learn to live with that. Thanks -

.

Relevant Pages

  • Re: Disturbed by alternating series.. help!
    ... it was pointed out to me that infinite alternating series do ... you are forgetting about absolute convergence. ... of finite sums, one can't expect infinite sums to behave the same way ...
    (sci.math)
  • Re: Need help with infinite series
    ... >>>It converges by the alternating series test. ... >>>It sums to approximately 0.6033175. ... take the average of two consecutive partial sums. ... you did we should get 8 decimals right instead ...
    (sci.math)