Convergence of infinite product to zero implies series divergence

Question

I am stuck in one step of an application of Borel Cantelli Lemma.

I am trying to show the following : if $a_n$ is a sequence such that $0 < a_n < 1$ for all $n$ then $\prod_{n=1}^\infty 1 - a_n = 0 \Rightarrow \sum_{n=1}^\infty a_n = +\infty$. By using the log I can show that $\prod_{n=1}^\infty 1 - a_n = 0 \Leftrightarrow \sum_{n=1}^\infty \log(1-a_n) = - \infty $ (since $1-a_n > 0, \forall n$) but I cannot conclude about the divergence of $\sum_{n=1}^\infty a_n$. Any hint ?

https://en.wikipedia.org/wiki/Infinite_product#Convergence_criteria — Theo Bendit, Mar 13 '18 at 16:55
If $a_n>\frac12$ infinitely often, $\sum a_n$ diverges. Thus one can assume that $0\leqslant a_n\leqslant\frac12$ for every $n$ and use the inequality $$e^{-2x}\leqslant1-x$$ for every $0\leqslant x\leqslant\frac12$. — Did, Mar 13 '18 at 16:55
Mmh I see, we use the equality then use the log ($-2a_n \leq log(1-a_n)$) and then sum over everything to conclude ? Thank you. — DimSum, Mar 13 '18 at 17:14
There are a few posts about this on this site, for example: Infinite product problem: $\sum p_n< \infty$ implies $\prod (1-p_n)>0$ or How to prove $\prod_{i=1}^{\infty} (1-a_n) = 0$ iff $\sum_{i=1}^{\infty} a_n = \infty$?, etc. — Martin Sleziak, Dec 15 '19 at 08:54

score 3 · Accepted Answer · answered Mar 13 '18 at 16:59

3

Hint:

If $\sum_{n=1}^\infty a_n < \infty$, given $\epsilon \in (0,1)$ there exists $N$ such that $\sum_{n=N}^\infty a_n < \epsilon$, and for all $m > N$

$$\prod_{n= N}^m(1-a_n) > 1 - \sum_{n=N}^ma_n > 1 - \epsilon > 0$$

answered Mar 13 '18 at 16:59

RRL

90,707

Ok I see that prove my point in that way $\prod_{n= 1}^\infty(1-a_n)$ cannot be zero but how do you prove $\prod_{n= N}^m(1-a_n) > 1 - \sum_{n=N}^ma_n$ ? – DimSum Mar 13 '18 at 17:07
3

@DimitriMeunier: $(1-a_N)(1-a_{N+1}) = 1 - (a_N + a_{N+1}) + a_N a_{N+1} > 1 - (a_N + a_{N+1})$. By induction $\prod_{n= N}^{m+1} (1- a_n) = \prod_{n= N}^{m} (1- a_n) (1 - a_{m+1}) > ( 1 - \sum_{n=N}^m a_n)(1 - a_{m+1}) > \ldots$. – RRL Mar 13 '18 at 17:22

Convergence of infinite product to zero implies series divergence

1 Answers1