1 × 1.1 × 1.01 × 1.0001 × 1.00000001 × 1.0000000000000001 × …
Zero anywhere sets the product to 0, regardless of the remaining factors. This seems a bit unruly, so we say that the product diverges if it approaches 0. Also, the product contains only positive factors. This avoids sign flipflopping, which is generally a bad idea if you want to approach a limit.
With these stipulations in place, take the log of each factor in the product to produce a corresponding series. Now partial products corespond to partial sums, and since log() is bicontinuous, the product converges iff the series converges. Since series terms approach 0, factors in the product approach 1. Also, the product is Cauchy, in a geometric sense. For every ε, factors in the product, beyond some point, have a ratio smaller than 1+ε.
Just as the sum of a linear combination of several series gives the linear combination of their limits, infinite products may be raised to various exponents and multiplied together. This corresponds to scaling and adding their logarithms.
If a product is bounded below some positive number, or above some ε > 0, and its terms are all greater than or less than 1 respectively, it is convergent. Switch to the log series and apply the analogous theorem.
A product absolutely converges if its log series does, and its factors may be permuted without changing the limit. Similarly, the factors of a conditionally convergent product can be rearranged to yield any positive limit.
A multidimensional product is well defined when its multidimensional log series is absolutely convergent, and one may multiply in row or column order. Take the log of everything and apply the analogous theorem.