On convergence of $\prod (1 - \alpha_n)$

Question

Suppose $\{ \alpha_n \}$ is a decreasing sequence of real numbers such that $0 < \alpha_n < 1$ and $\alpha_n$ goes to $0$ as $n$ goes to infinity.

I was wondering if there is a known condition for $\{ \alpha_n \}$ so that the product $\prod (1- \alpha_n)$ will not be $0$?

Thanks!

Answer 1

Everytime when the Product converges the limit won't be zero, because per definition a infinite product only converges when it limit is not zero. Using that

$$\log\Big( \prod_{i=1}^n a_i \Big) = \sum_{i=1}^n \log(a_i) $$ one just can use the well known results for series to test the convergence of products.