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?