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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.

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    $\begingroup$ Well, one example is to compute probabilities associated with an infinite sequence of independent coin flips. $\endgroup$ – Michael Jun 5 '16 at 3:39
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    $\begingroup$ Along these lines, the Borel-Cantelli lemma. $\endgroup$ – Michael Jun 5 '16 at 3:40
  • $\begingroup$ Sometimes it’s just more natural to approximate a quantity or object by products than by sums. My favorite proof of Weierstrass Preparation gets the desired objects by the infinite product process, for instance. $\endgroup$ – Lubin Jun 5 '16 at 3:42
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    $\begingroup$ See also Euler products in number theory. $\endgroup$ – carmichael561 Jun 5 '16 at 3:43
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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 $\sum_{n=1}^{\infty}(1/p_n)=\infty.$

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An example in the economics of natural resources. Suppose there are limited reserves of a non-renewable resource, eg a mineral, and we want to use dynamic optimisation with an infinite horizon to identify the optimal time path for extraction and use of the resource. ('Optimal' might for example be defined in terms of maximising the discounted present value of consumption of goods produced using the resource together with other resources such as capital and labour.)

If we work in discrete periods, eg years, it may be convenient to define $S_t$ as the stock of the resource remaining at the start of period $t$, and $r_t$ as the proportion of $S_t$ extracted and used in period $t$. Thus:

$$S_{t+1}=S_t(1-r_t)$$

Maximisation will normally require:

$$\lim_{t\to\infty}S_t=0$$

Hence we will want to impose the following condition in terms of $r_t$:

$$\prod_{t=1}^{\infty}(1-r_t)=0$$

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