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 $\sum_{n=1}^\infty a_n=\infty$.

Question

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 $\sum_{n=1}^\infty a_n=\infty$.

Proof of $\Rightarrow$: Assume that $\prod_{n=1}^\infty (1-a_n)=0$ converges. Well then at some $i$, $1-a_i=0$, which implies $a_i=1$. Knowing this if we were to $$\sum_{n=1}^\infty a_n=\underbrace{1+\dots+1}_{\infty \text{ times}}=1\cdot \infty=\infty.$$

Proof of $\Leftarrow$: Assume $\sum_{n=1}^\infty a_n=\infty$. How would I do this direction?

your proof is incorrect. You need to use the defintion of limit to prove it. — runaround, Feb 11 '16 at 06:50
Did you have a look at frequent questions about infinite product? These two questions seem related, maybe you will find more: http://math.stackexchange.com/questions/519656/how-to-prove-prod-i-1-infty-1-a-n-0-iff-sum-i-1-infty-a-n and http://math.stackexchange.com/questions/380094/sufficiency-and-necessity-of-convergence-of-sum-a-n-wrt-convergence-of-prod — Martin Sleziak, Feb 11 '16 at 07:32
BTW this seem to be closely related to your earlier questions: http://math.stackexchange.com/questions/1649878/suppose-a-n0-for-n-in-mathbbn-prove-that-prod-n-1-infty-1a-n-c and http://math.stackexchange.com/questions/1649989/suppose-1a-n0-for-n-in-mathbbn-prove-that-prod-n-1-infty-1-a-n — Martin Sleziak, Feb 11 '16 at 07:38

score 3 · Accepted Answer · answered Feb 11 '16 at 07:31

The convention for infinite products is to say that the infinite product diverges to $0$ when the limit of partial products is $0$.

To get you on the correct track, here is one side of a correct proof. Start with the well-known inequality $1-x \leqslant e^{-x}$ for $0 \leqslant x < 1$.

With $ 0 < a_n < 1$, it follows that for any $m \in \mathbb{N}$,

$$0 < P_m = \prod_{n=1}^m(1-a_n) \leqslant \prod_{n=1}^m e^{-a_n} = \exp\left(-\sum_{n=1}^ma_n\right).$$

Hence if $\sum_{n=1}^{\infty}a_n = \infty$ then $\lim_{m \to \infty} P_m = 0.$

score 3 · Answer 2 · answered Feb 11 '16 at 08:12

For the other direction: There is an easy generalization of Bernoulli's inequality that reads $$ \prod_{n=N}^\infty (1-a_n)\ge 1-\sum_{n=N}^\infty a_n $$ so if the series converges one can find an $N$ such that the remainder is smaller than $\sum_{n=N}^\infty a_n<\frac12$ and thus $$ \prod_{n=1}^\infty (1-a_n)\ge\frac12\prod_{n=1}^{N-1} (1-a_n)>0. $$

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 $\sum_{n=1}^\infty a_n=\infty$.

2 Answers2

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