Definition:Convergent Product
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Definition
Convergent Product in a Standard Number Field
Let $\sequence {a_n}$ be a sequence of nonzero elements of $\mathbb K$.
Then:
- The infinite product $\ds \prod_{n \mathop = 1}^\infty a_n$ is convergent
- its sequence of partial products converges to a nonzero limit $a \in \mathbb K \setminus \set 0$.
Convergent Product in Arbitrary Field
Let $\sequence {a_n}$ be a sequence of nonzero elements of $\mathbb K$.
The infinite product $\ds \prod_{n \mathop = 1}^\infty a_n$ is convergent if and only if its sequence of partial products converges to a nonzero limit $a \in \mathbb K \setminus \set 0$.
Convergent Product in Normed Algebra
The infinite product $\ds \prod_{n \mathop = 1}^\infty a_n$ is convergent if and only if there exists $n_0\in\N$ such that:
- $(1): \quad a_n$ is invertible for $n \ge n_0$
- $(2): \quad$ the sequence of partial products of $\ds \prod_{n \mathop = n_0}^\infty a_n$ converges to some invertible $b\in A^\times$.
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 $\ds \prod_{n \mathop = n_0 + 1}^\infty a_n$ converges to $0$
the product diverges to $0$, and we assign the value:
- $\ds \prod_{n \mathop = 1}^\infty a_n = 0$
Informal Definition
A convergent product is an infinite (continued) product whose sequence of partial products converges.
Also see
- Logarithm of Infinite Product of Real Numbers
- Logarithm of Infinite Product of Complex Numbers
- Definition:Absolute Convergence of Product
- Definition:Cauchy's Criterion for Products
- Results about convergent products can be found here.
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): convergent product
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): convergent product