= Positive sum-product convergence criterion
For $a_j\geq0$, the finite product satisfies
$$
1+\sum_{j=1}^na_j\leq\prod_{j=1}^n(1+a_j)\leq\exp\left(\sum_{j=1}^na_j\right).
$$
The lower inequality follows by product expansion; the upper one uses $1+t\leq e^t$. Since both the partial sums and partial products are increasing, one converges to a finite limit exactly when the other does. The <monotone bounded sequence> theorem proves both implications. The product cannot tend to zero in this nonnegative setting.
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