# Sequences and Series/Infinite products

Definition (infinite product):

Let ${\displaystyle (b_{n})_{n\in \mathbb {N} }}$ be a sequence of numbers in ${\displaystyle \mathbb {K} =\mathbb {R} }$ or ${\displaystyle \mathbb {C} }$. If the limit

${\displaystyle \lim _{N\to \infty }\prod _{n=1}^{N}b_{n}}$

exists, it is called the infinite product of ${\displaystyle (b_{n})_{n\in \mathbb {N} }}$ and denoted by

${\displaystyle \prod _{n=1}^{\infty }b_{n}}$.

Proposition (necessary condition for convergence of infinite products):

In order for the infinite product

${\displaystyle \prod _{n=1}^{\infty }b_{n}}$

of a sequence ${\displaystyle (b_{n})_{n\in \mathbb {N} }}$ to exist and not to be zero, it is necessary that

${\displaystyle \lim _{n\to \infty }b_{n}=1}$.

Proof: Suppose that not ${\displaystyle \lim _{n\to \infty }b_{n}=1}$. Then there exists ${\displaystyle \epsilon >0}$ and an infinite sequence ${\displaystyle (n_{k})_{k\in \mathbb {N} }}$ such that for all ${\displaystyle k\in \mathbb {N} }$ we have ${\displaystyle |b_{n}-1|>\epsilon }$. Thus, upon denoting

${\displaystyle P_{N}:=\prod _{n=1}^{N}b_{n}}$,

we will have

${\displaystyle |P_{n_{k}}-P_{n_{k}-1}|=|P_{n_{k}-1}||b_{n}|}$.

Suppose for a contradiction that ${\displaystyle \lim _{N\to \infty }P_{N}}$ exited and was equal to ${\displaystyle c\in \mathbb {R} }$. Then when ${\displaystyle k}$ is sufficiently large, we will have

${\displaystyle \left||P_{n_{k}-1}||b_{n}|-|c|\right|\geq |c|\epsilon /2}$,

which is a contradiction. ${\displaystyle \Box }$

Proposition (series criterion for the convergence of infinite products):

Let ${\displaystyle (a_{n})_{n\in \mathbb {N} }}$ be a sequence of real numbers. If

${\displaystyle \sum _{n=1}^{\infty }|a_{n}|<\infty }$,

then

${\displaystyle \prod _{n=1}^{\infty }(1+a_{n})}$

converges.

Proof: ${\displaystyle \Box }$