Ratio test for infinite products?

Question

Is the ratio test applicable for testing convergence of infinite products?

In other words, consider the sequence $(a_i)_{i=1}^\infty$ of non-zero real numbers.

Also, consider the product $\displaystyle\mathcal P=\prod_{k=1}^\infty a_k$

Are the following statements true?

$$\lim_{n\to\infty}\left|\frac{a_{k+1}}{a_k}\right|\lt 1\implies \mathcal P\textrm{ converges}\\ \lim_{n\to\infty}\left|\frac{a_{k+1}}{a_k}\right|\gt 1\implies \mathcal P\textrm{ diverges}$$

Answer 1

With products, $a_k\to 1$ is the right assumption, otherwise stupid things happen. Just as in series, we usually just assume $a_k \to 0$. So in this context, $$ \lim_{k\to\infty}\left|\frac{a_{k+1}}{a_{k}}\right| =1 $$ should happen every time.

It's worth noting that $$\prod_{k=1}^{\infty}a_k$$ converges iff $$\sum_{i=1}^k\log(a_k)$$

Think about the exponential function to see why this is. So really, what you want to test is

$$ \lim_{k\to\infty}\left|\frac{\log(a_{k+1})}{\log(a_{k})}\right| $$

Answer 2

A simple way to test if an infinite product converges or diverges is to test the related sum:

$$P=\prod_{k=1}^\infty {(a_k)}$$

can be turned into:

$$\ln{(P)}=S = \sum_{k=1}^\infty {(\ln{a_k})}$$

Now, the product $P$ converges if the sum $S$ converges. If $S$ diverges, then $P$ diverges as well.

You can now use the appropriate test on $S$ (whether it be $n$th term test, ratio test, etc.).