Complex infinite-product convention extending the published real definition

The published definition Infinite products: partial products, and convergence to a nonzero limit after finitely many vanishing factors is stated for real factors. This page extends its tail convention explicitly to complex factors: for a complex sequence $(b_n)$, the product ${\prod }_{n\ge 0}{b}_n$ converges when there is an index $N$ such that $b_n\mathrm{\ne }0$ for every $n\ge N$ and the complex partial products ${\prod }_{n=N}^m{b}_n$ converge as $m\to \mathrm{\infty }$ to a nonzero complex limit.

For a complex sequence $(a_n)$, the phrase

${\prod }_{n\ge 0}(1+a_n)\text{ converges absolutely}$

means exactly that the real product

${\prod }_{n\ge 0}(1+\mathrm{\mid }{a}_n\mathrm{\mid })$

converges in the sense of Infinite products: partial products, and convergence to a nonzero limit after finitely many vanishing factors. By For $p_k\ge 0$ the product $\prod (1+p_k)$ converges iff $\sum p_k$ converges, with $1+{\sum }_{k<n}{p}_k\le {\prod }_{k<n}(1+p_k)\le 1\mathrm{/}(1-{\sum }_{k<n}{p}_k)$ when ${\sum }_{k<n}{p}_k<1$; for $0\le p_k<1$ the product $\prod (1-p_k)$ converges iff $\sum p_k$ converges and its partial products tend to $0$ otherwise; and $\sum \mathrm{\mid }{p}_k\mathrm{\mid }$ convergent implies $\prod (1+p_k)$ convergent, this is equivalent to the numerical series ${\sum }_{n\ge 0}\mathrm{\mid }{a}_n\mathrm{\mid }$ converging.

The zero-factor convention is also unchanged. A finite number of zero factors is harmless because convergence is tail-based, but infinitely many zero factors prevent any admissible nonzero tail limit. Later theorems will therefore isolate finite zero sets first and then work on zero-free tails.