Convergent series

Erken

Let $\{a_n\}$ be a sequence with $a_n\in (0,1)$ for all $n$ . Prove that the infinite product
$\prod_{n = 1}^{\infty}(1 - a_n) = \lim_{n\rightarrow\infty} \prod_{i = 1}^{n}(1 - a_i)$
is positive if and only if the series $\sum_{i = 1}^{\infty}a_i$ is convergent.

**Posted:** Sun Nov 08, 2009 3:06 am

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A mathematician is a scientist who can figure out anything except such simple things as squaring the circle and trisecting an angle

Diogene

Moderator says: it never makes sense to quote the preceding post in full (especially, if it is the only other post in the thread)
$0 < \prod_{n = 1}^{\infty}(1 - a_n) < 1 \ \Longleftrightarrow \ \ - \infty < \sum_{n = 1}^{\infty}ln(1 - a_n) < 0$ . But $ln(1 - a_n) < - a_n < 0$ and it's easy to prove the the requirement :
$- \infty < \sum_{n = 1}^{\infty}ln(1 - a_n) \Longrightarrow 0 < \sum_{n = 1}^{\infty}a_n < \infty$ , and, $\sum_{n = 1}^{\infty}a_n < \infty \Longrightarrow - \infty < \sum_{n = 1}^{\infty}ln(1 - a_n)$
Cool

**Posted:** Sun Nov 08, 2009 10:24 am

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D.. to A.. : "Stand out of my light."

Diogene

Moderator, I don't agree Mr. Green

. It makes sense, because some times the "preceding post" can be modified by its autor .. Cool

**Posted:** Sun Nov 08, 2009 2:03 pm

_________________
D.. to A.. : "Stand out of my light."