Sequences and Series/Infinite products

Definition (infinite product):

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

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

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

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

Proposition (necessary condition for convergence of infinite products):

In order for the infinite product

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

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

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

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

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

we will have

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

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

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

which is a contradiction. $\Box$

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

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

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

then

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

converges.

Proof: $\Box$