Convergence of series of elementary symmetric functions

Question

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?

Answer 1

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.