Convergence and absolute convergence of an infinite product of terms in $(0,1]$.

Question

Let $x_i\in(0,1]$ for all $i$. Are the following true, and if so is there a easy proof or citation.

1. $\prod_{i=1}^\infty x_i = e^{\sum_{i=1}^\infty \log x_i}$ always holds (if the sum diverges to $-\infty$, the equality is $0=0$).

2. If $\sum_{i=1}^\infty |\log x_i|<\infty$, then $\prod_{i=1}^\infty x_i$ converges absolutely (in the sense that the value doesn't change with reordering).

For 2, if that's not true, is there some condition about absolute convergence of a series that implies absolute convergence of the infinite product.

Answer 1

Both are true.

1. Follows by letting $N \to \infty$ in $\prod_{i=1}^{N} x_i=e^{ \sum\limits_{i=1}^{N} \log x_i}$

2. follows immediately from 1) and the fact that if series is absolutely convergent then any permutation of the terms results in a convergent series.