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In mathematics, an iterated product (or simply a product) is the result of repeatedly applying the binary operation of multiplication to a sequence of elements.^[1]^[2]

The factors being multiplied may be integers, real numbers, complex numbers, matrices, polynomials, functions, or, more generally, elements of a monoid equipped with a multiplication operation. For finite sequences, the iterated product always yields a well-defined result.^[3]

When the sequence contains infinitely many factors, the corresponding construction is known as an infinite product. In this case, the value of the product is defined using the concept of a limit and does not necessarily exist.^[4]

Notation

Product notation

The product of a finite sequence is commonly written using the capital Greek letter pi, $\prod$ :

\prod _{k=m}^{n}x_{k}=x_{m}x_{m+1}x_{m+2}\cdots x_{n},

where $k$ is the index of the product, $m$ is the lower bound, and $n$ is the upper bound.^[5]

For example,

\prod _{k=2}^{5}k=2\cdot 3\cdot 4\cdot 5=120.

^[3]

More general forms include:

\prod _{0\leq k<100}f(k),

which denotes the product of all values of $f(k)$ satisfying the stated condition, and

\prod _{x\in S}f(x),

which denotes the product of $f(x)$ over all elements of a set $S$ .^[3]

Multiple products may be written as

\prod _{k}\prod _{l}a_{kl}

or equivalently

\prod _{k,l}a_{kl}.

Special cases

Product notation can also be applied to fewer than two factors:

The product of a single factor $x$ is $x$ .
The product of no factors is defined to be $1$ , the multiplicative identity. This is known as the empty product.^[6]

Consequently,

if $m=n$ , the product contains exactly one factor and equals $x_{m}$ ;
if $m>n$ , the product is empty and equals $1$ .

Identities

The following identities hold for finite products:^[7]

$\prod _{k=m}^{n}(rf(k))=r^{n-m+1}\prod _{k=m}^{n}f(k)$
$\prod _{k=m}^{n}f(k)g(k)=\left(\prod _{k=m}^{n}f(k)\right)\left(\prod _{k=m}^{n}g(k)\right)$
$\prod _{k=m}^{n}{\frac {f(k)}{g(k)}}={\frac {\prod _{k=m}^{n}f(k)}{\prod _{k=m}^{n}g(k)}}$
$\prod _{k=m}^{n}f(k)=\prod _{k=m+r}^{n+r}f(k-r)$
$\prod _{k=m}^{r}f(k)\prod _{k=r+1}^{n}f(k)=\prod _{k=m}^{n}f(k)$
$\ln !\left(\prod _{k=m}^{n}f(k)\right)=\sum _{k=m}^{n}\ln f(k)$
$\prod _{k=1}^{n}k=n!$
$\prod _{k=m}^{n}k={\frac {n!}{(m-1)!}}$
$\prod _{k=m}^{n}r=r^{,n-m+1}$
$\prod _{k=1}^{n}{\frac {r-k+1}{k}}={\binom {r}{n}}$
$\prod _{k=m}^{n}{\frac {k+1}{k}}={\frac {n+1}{m}}$ (a telescoping product)
$\prod _{k=0}^{\lceil n/2\rceil -1}(n-2k)=n!!$
$\prod _{k=0}^{n}r^{k}=r^{\sum _{k=0}^{n}k}=r^{n(n+1)/2}$
$\prod _{k=0}^{n}r^{f(k)}=r^{\sum _{k=0}^{n}f(k)}$

References

^ Rodda, Harvey J. E. (2015). Understanding Mathematical and Statistical Techniques in Hydrology: An Examples-Based Approach. John Wiley & Sons. p. 41. ISBN 978-1-4443-3549-1.
^ Cuninghame-Green, Raymond A. (1979). Minimax Algebra. Lecture Notes in Economics and Mathematical Systems. Springer-Verlag. p. 7. ISBN 978-3-642-48708-8.
^ ^a ^b ^c “Pi Notation (Product Notation)”. MathMaine. 4 March 2018.
^ Weisstein, Eric W. “Infinite Product”. MathWorld.
^ Cuninghame-Green, Raymond A. (1979). Minimax Algebra. Springer-Verlag. p. 7.
^ Linear Algebra and Geometry.
^ Weisstein, Eric W. “Product”. MathWorld.

[rdp-we-cite_note-1] Rodda, Harvey J. E. (2015). Understanding Mathematical and Statistical Techniques in Hydrology: An Examples-Based Approach. John Wiley & Sons. p. 41. ISBN 978-1-4443-3549-1.

[rdp-we-cite_note-2] Cuninghame-Green, Raymond A. (1979). Minimax Algebra. Lecture Notes in Economics and Mathematical Systems. Springer-Verlag. p. 7. ISBN 978-3-642-48708-8.

[rdp-we-cite_note-:0-3] “Pi Notation (Product Notation)”. MathMaine. 4 March 2018.

[rdp-we-cite_note-4] Weisstein, Eric W. “Infinite Product”. MathWorld.

[rdp-we-cite_note-CG-5] Cuninghame-Green, Raymond A. (1979). Minimax Algebra. Springer-Verlag. p. 7.

[rdp-we-cite_note-6] Linear Algebra and Geometry.

[rdp-we-cite_note-7] Weisstein, Eric W. “Product”. MathWorld.

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Sample Page

Notation

Product notation

Special cases

Identities

See also

References