Definition:Continued Product
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Definition
Let $\struct {S, \times}$ be an algebraic structure where the operation $\times$ is an operation derived from, or arising from, the multiplication operation on the natural numbers.
Let $\tuple {a_1, a_2, \ldots, a_n} \in S^n$ be an ordered $n$-tuple in $S$.
Definition by Index
The composite is called the continued product of $\tuple {a_1, a_2, \ldots, a_n}$, and is written:
- $\ds \prod_{j \mathop = 1}^n a_j = \paren {a_1 \times a_2 \times \cdots \times a_n}$
Definition by Inequality
The continued product of $\tuple {a_1, a_2, \ldots, a_n}$ can be written:
- $\ds \prod_{1 \mathop \le j \mathop \le n} a_j = \paren {a_1 \times a_2 \times \cdots \times a_n}$
Definition by Propositional Function
Let $\map R j$ be a propositional function of $j$.
Then we can write:
- $\ds \prod_{\map R j} a_j = \text { the product of all $a_j$ such that $\map R j$ holds}$.
If more than one propositional function is written under the product sign, they must all hold.
Infinite
Let an infinite number of values of $j$ satisfy the propositional function $\map R j$.
Then the precise meaning of $\ds \prod_{\map R j} a_j$ is:
- $\ds \prod_{\map R j} a_j = \paren {\lim_{n \mathop \to \infty} \prod_{\substack {\map R j \\ -n \mathop \le j \mathop < 0} } a_j} \times \paren {\lim_{n \mathop \to \infty} \prod_{\substack {\map R j \\ 0 \mathop \le j \mathop \le n} } a_j}$
provided that both limits exist.
If either limit does fail to exist, then the infinite product does not exist.
Index Variable
Consider the continued product, in either of the three forms:
- $\ds \prod_{j \mathop = 1}^n a_j \qquad \prod_{1 \mathop \le j \mathop \le n} a_j \qquad \prod_{\map R j} a_j$
The variable $j$, an example of a bound variable, is known as the index variable of the continued product.
Multiplicand
The set of elements $\set {a_j \in S}$ is called the multiplicand.
Notation
The sign $\ds \prod$ is called the product sign and is derived from the capital Greek letter $\Pi$, which is $\mathrm P$, the first letter of product.
Vacuous Product
Take the composite expressed as a continued product:
- $\ds \prod_{\map R j} a_j$
where $\map R j$ is a propositional function of $j$.
Suppose that there are no values of $j$ for which $\map R j$ is true.
Then $\ds \prod_{\map R j} a_j$ is defined to be $1$.
Beware: not zero.
This composite is called a vacuous product.
This is because:
- $\forall a: a \times 1 = a$
where $a$ is a number.
Hence for all $j$ for which $\map R j$ is false, the value of the product is unaffected.
This is most frequently seen in the form:
- $\ds \prod_{j \mathop = m}^n a_j = 1$
where $m > n$.
In this case, $j$ can not at the same time be both greater than or equal to $m$ and less than or equal to $n$.
Also known as
A continued product can also be seen as product notation, but such a term is not only imprecise but also ambiguous.
Also see
- Results about continued products can be found here.
Historical Note
The originally investigation into the theory of infinite products was carried out by Leonhard Paul Euler.
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): continued product
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): continued product
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): product notation