My analysis book covers a section on infinite products. So I started wondering what the practical applications of infinite products are in science and engineering, but couldn't find anything yet. Also, what are common applications in pure math?
Thanks for the info.
One example: Infinite series occur in just about every branch of applied math, and it is necessary to have tests for convergence of them.
Theorem 1. If $a_n\geq 0$ for $n\in N,$ then $$\sum_{n=1}^{\infty}a_n<\infty \iff \prod_{n=1}^{\infty}(1+a_n)<\infty.$$
Example: Let $a_n=1/n.$ Then $\prod_{n=1}^m(1+a_n)=m+1,$ which $\to \infty$ as $m\to \infty.$ Therefore $\sum_{n=1}^{\infty}(1/n)=\infty.$
Theorem 2. If $0\leq a_n<1$ for $n\in N$ then $$\sum_{n=1}^{\infty}a_n<\infty \iff \prod_{n=1}^{\infty}(1-a_n)>0.$$
Euler used this for a new way of showing that there are infinitely many primes, along with a new result: Let $p_n$ be the $n$-th prime . Then $\sum_{n=1}^{\infty}(1/p_n)=\infty.$