# Convergence of Infinite products

References

Theorem

Let $\{a_n\}$ be a sequence of positive numbers. Then the infinite product

(1)     $\prod_{n=1}^{\infty}(1 + a_n)$

converges if and only if the series

(2)     $\sum_{n=1}^{\infty}a_n$

converges.

Proof

We first take the logarithm of (1), we have the series

(3)     $\sum_{n=1}^{\infty}log(1+a_n)$

and if (2) converges, then

$\lim_{n\to\infty}a_n=0$

So, we can assume that $a_n\to 0$ as $n\to\infty$.

Since we have

(4)     $\lim_{x\to 0}\frac{log(x+1)}{x} = \lim_{x\to 0}\frac{log'(x+1)}{x'}$  (L’Hospital)

$=\lim_{x\to 0}\frac{1}{x+1}=1 > 0$

Using the Limit Comparison test, if (2) converges $\Leftrightarrow$ (3) converges$\Leftrightarrow$ (1) converges. Q.E.D.