Convergence of Infinite products

References

[1] The everything of seminar

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.

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