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Let $\{p_k\}$ be a probability mass sequence. Is it true that if $\prod_{k=0}^\infty p_k > 0$ then $\sum_{k=0}^\infty (1-p_k) < \infty$?

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Actually, I just noticed that my question does not make sense, because if $\{p_k\}$ is a probability mass sequence, then $\sum_{k=0}^\infty p_k = 1$, but then $\prod_{k=0}^\infty p_k$ cannot be greater than $0$. I'm sorry. – user67398 Mar 20 '13 at 1:59

See, especially the section "Convergence criteria".

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