When will an infinite product converge?

Question

By crude intuition, it seems that an infinite product would either diverge, or converge to $1$. Do infinite products converge to other values? Is testing for such convergence similar to the tests applied to infinite sums?

Answer 1

If the numbers are positive, then

$$\log(a_1\cdot a_2\cdot a_3\cdots) = \log(a_1) + \log(a_2) + \cdots $$

so the product converges if and only if the sum of the logarithms converges, and

$$\prod_{i=1}^\infty a_i = e^{\sum_{i=1}^\infty \log(a_i)}$$

Answer 2

Example (Wallis' product):

$ \prod_{k=1}^{\infty }\frac{4k^{2}}{4k^{2}-1}={\frac {\pi }{2}} \ne 1.$

Answer 3

For an infinite product to converge, the terms must tend to $1$, just as the terms of a convergent infinite sum must tend to $0$, but the product itself can be anything. For example,

$$ \begin{align} \prod_{k=2}^\infty\frac{k^3+1}{k^3-1} &=\lim_{n\to\infty}\prod_{k=2}^n\color{#C00}{\frac{k+1}{k-1}}\color{#090}{\frac{k^2-k+1}{k^2+k+1}}\\ &=\lim_{n\to\infty}\color{#C00}{\frac{n(n+1)}2}\color{#090}{\frac3{n^2+n+1}}\\[3pt] &=\frac32 \end{align} $$ whereas $$ \lim_{k\to\infty}\frac{k^3+1}{k^3-1}=1 $$