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Let $0 \leq f(x), g(x,y) < b$ be two continuous real-valued functions where $b \in \mathbb{R}^+$. Is there a way to rearrange the following infinite sum of finite products into a finite product of infinite sums?

$$\sum_{i=0}^{\infty} f(i)\prod_{j=0}^{n} g(i,j)$$

In a sense, I see this as a sort of generalization to this answer, but this one suggests to me that some special handling may be possible when infinite series are at play.

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