Definition:Convergent Product

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Convergent Product in a Standard Number Field

Let $\mathbb K$ be one of the standard number fields $\Q, \R, \C$.

Nonzero Sequence

Let $\sequence{a_n}$ be a sequence of nonzero elements of $\mathbb K$.


The infinite product $\displaystyle \prod_{n \mathop = 1}^\infty a_n$ is convergent if and only if its sequence of partial products converges to a nonzero limit $a\in\mathbb K\setminus\{0\}$.


Arbitrary Sequence

Let $\sequence{a_n}$ be a sequence of elements of $\mathbb K$.


The infinite product $\displaystyle \prod_{n \mathop = 1}^\infty a_n$ is convergent if and only if:

The sequence of partial products of $\displaystyle \prod_{n \mathop = 1}^\infty a_n$ is then convergent to some $a\in\mathbb K$.


Convergent Product in Arbitrary Field

Let $\struct {\mathbb K, \norm {\,\cdot\,} }$ be a valued field.

Nonzero Sequence

Let $\sequence {a_n}$ be a sequence of nonzero elements of $\mathbb K$.


The infinite product $\displaystyle \prod_{n \mathop = 1}^\infty a_n$ is convergent if and only if its sequence of partial products converges to a nonzero limit $a \in \mathbb K \setminus \set 0$.


Arbitrary Sequence

Let $\sequence {a_n}$ be a sequence of elements of $\mathbb K$.


The infinite product $\displaystyle \prod_{n \mathop = 1}^\infty a_n$ is convergent if and only if:

There exists $n_0 \in \N$ such that the sequence of partial products of $\displaystyle \prod_{n \mathop = n_0}^\infty a_n$ converges to some $b \in \mathbb K \setminus \set 0$.

The sequence of partial products of $\displaystyle \prod_{n \mathop = 1}^\infty a_n$ is then convergent to some $a \in \mathbb K$.


Convergent Product in Normed Algebra

Let $\mathbb K$ be a division ring with norm $\norm{\,\cdot\,}_{\mathbb K}$.

Let $\struct{ A, \norm{\,\cdot\,} }$ be an associative normed unital algebra over $\mathbb K$.

Let $\sequence{a_n}$ be a sequence in $A$.


Definition 1

The infinite product $\displaystyle \prod_{n \mathop = 1}^\infty a_n$ is convergent if and only if there exists $n_0\in\N$ such that:

  1. $a_n$ is invertible for $n \geq n_0$
  2. the sequence of partial products of $\displaystyle \prod_{n \mathop = n_0}^\infty a_n$ converges to some invertible $b\in A^\times$.


Definition 2: for complete algebras

Let $\struct{ A, \norm{\,\cdot\,} }$ be complete.


The infinite product $\displaystyle \prod_{n \mathop = 1}^\infty a_n$ is convergent if and only if there exists $n_0\in\N$ such that:

the sequence of partial products of $\displaystyle \prod_{n \mathop = n_0}^\infty a_n$ converges to some invertible $a\in A^\times$.


Divergent Product

An infinite product which is not convergent is divergent.


Divergence to zero

If either:

there exist infinitely many $n \in \N$ with $a_n = 0$
there exists $n_0 \in \N$ with $a_n \ne 0$ for all $n > n_0$ and the sequence of partial products of $\displaystyle \prod_{n \mathop = n_0 + 1}^\infty a_n$ converges to $0$

the product diverges to $0$, and we assign the value:

$\displaystyle \prod_{n \mathop = 1}^\infty a_n = 0$


Also defined as

Some authors define an infinite product to be convergent if and only if the sequence of partial products converges. The phrase:

$\displaystyle \prod_{n \mathop = 1}^\infty a_n$ diverges to $0$

then becomes:

$\displaystyle \prod_{n \mathop = 1}^\infty a_n$ converges to $0$.

The definition used here has many desirable consequences, such as:

Factors in Convergent Product Converge to One
Convergence of Infinite Product Does not Depend on Finite Number of Factors
Product of Convergent and Divergent Product is Divergent

which creates an analogy with series.


Also see