# Convergence of series of elementary symmetric functions

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

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.
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