Iterated product 📖 Wikipedia

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]

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

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

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

For example,

${\displaystyle \prod _{k=2}^{5}k=2\cdot 3\cdot 4\cdot 5=120.}$^[3]

More general forms include:

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

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

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

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

Multiple products may be written as

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

or equivalently

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

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

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

Consequently,

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

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

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

• Product (mathematics)
• Infinite product
• Empty product
• Factorial
• Monoid