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Let $x_1,x_2,x_3,\ldots$ be an infinite sequence of real numbers (or assume they're complex numbers if you find that convenient).

Let $e_0,e_1,e_2,e_3,\ldots$ be the elementary symmetric functions of $x_1,x_2,x_3,\ldots$ .

If I'm not mistaken, the two series \begin{align} & e_0-e_2+e_4-\cdots \\[6pt] & e_1-e_3+e_5-\cdots \end{align} converge absolutely if $\displaystyle\sum_{j=1}^\infty x_j$ converges absolutely.

So:

  • What proofs of this are known and where are they?
  • Or, if I'm mistaken, what's a counterexample?
  • Can anything sensible be said about conditional convergence?
  • Has anything been said about conditional convergence in refereed publications?
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I added a "combinatorics" tag on the theory that that would attract the attention of some people who might know about this. –  Michael Hardy Nov 2 '12 at 17:08
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up vote 3 down vote accepted

Note that $|e_n|$ is dominated by $\left(\sum_j|x_j|\right)^n/(n!)$, so yes, everything like what you wrote converges and pretty fast.

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Is this mentioned in authoritative sources that I can cite? –  Michael Hardy Nov 2 '12 at 18:26
    
I can be pretty authoritative myself if that's what you need :). Honestly, I have no idea. Probably yes, but if I were writing a paper and needed this fact, I would just include the line I wrote as an answer and not bother about any citations. Indeed, what's the point of sending the reader to the library if you can tell the whole story in one line? –  fedja Nov 3 '12 at 0:13
    
What I might do is write a paper in which I mention in passing the fact that if $x_1+x_2+x_3+\cdots$ converges absolutely then so do the two series I asked about, but that I prefer not to rely on that, but instead to do something else. So a one-line comment might be just what I need. –  Michael Hardy Nov 3 '12 at 2:45
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