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

Question

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

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

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.

Kavi Rama Murthy · Answer

Both are true.

Follows by letting $N to infty$ in $prod_{i=1}^{N} x_i=e^{ sumlimits_{i=1}^{N} log x_i}$

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

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