If $\displaystyle\prod_{n=1}^{\infty} (1-a_{n}) = 0$ then is it always true that $\displaystyle\sum_{n=1}^{\infty} a_{n}$ diverges? ($0 \leq a_{n} < 1) $
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See also: Infinite product problem: $\sum p_n< \infty$ implies $\prod (1-p_n)>0$, Suppose $1>a_n>0$ for $n\in \mathbb{N}$. Prove that $\prod_{n=1}^\infty (1-a_n)=0$ converges if and only if $\sum_{n=1}^\infty a_n=\infty$., How to prove $\prod_{i=1}^{\infty} (1-a_n) = 0$ iff $\sum_{i=1}^{\infty} a_n = \infty$?, etc. – Martin Sleziak Dec 16 '19 at 16:26
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If $a_n$ does not approach $0$ as $n \to \infty$, then $\displaystyle\sum_{n = 1}^{\infty}a_n$ diverges. So, assume $\displaystyle\lim_{n \to \infty}a_n = 0$
If $\displaystyle\prod_{n = 1}^{\infty}(1-a_n) = 0$, then $\displaystyle\sum_{n = 1}^{\infty}-\ln(1-a_n) = \infty$. Now, use the limit comparison test.
JimmyK4542
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(+1) ... or just squeezing, since $a_n\geq -\frac{1}{2}\log(1-a_n)$ for every $a_n$ sufficiently close to zero. – Jack D'Aurizio Sep 11 '14 at 02:30