Disturbed by alternating series.. help!


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?

.

Relevant Pages

  • Re: Disturbed by alternating series.. help!
    ... > I was under the mistaken impression that the re-ordering of a sum could ... But the sum of an infinite series is not an ordinary sum. ... it was pointed out to me that infinite alternating series do ...
    (sci.math)
  • Re: Disturbed by alternating series.. help!
    ... i.e. the commutative property of addition. ... it was pointed out to me that infinite alternating series ... Don't confuse a sum with the limit of sums. ... Unsolicited bulk E-mail subject to legal action. ...
    (sci.math)
  • Re: cosine
    ... If L is the sum of an alternating series and s_n are the partial ... the job for this limited range of x. ... If you're allowed to use the periodicity of cos x, ...
    (sci.math)
  • Re: Need help with infinite series
    ... >>It converges by the alternating series test. ... >(You took the sum of the first 100,000 terms. ... we get that the original series sums to 0.604898643421630370247... ...
    (sci.math)
  • Re: Why is it ?
    ... I have never seen it used for alternating series. ... would express the alternating sum from n to infinity as ... are those of the Statistics Department or of Purdue University. ... Herman Rubin, Department of Statistics, Purdue University ...
    (sci.math)
Quantcast