Wallis product
English
Etymology
Published in 1656 by John Wallis.
Proper noun
the
Wallis
product
(
mathematics
)
An
infinite
product
that
approximates
half of
pi
:
π
2
=
∏
n
=
1
∞
4
n
2
4
n
2
−
1
=
∏
n
=
1
∞
(
2
n
2
n
−
1
⋅
2
n
2
n
+
1
)
=
(
2
1
⋅
2
3
)
⋅
(
4
3
⋅
4
5
)
⋅
(
6
5
⋅
6
7
)
⋅
(
8
7
⋅
8
9
)
⋅
⋯
{\displaystyle {\begin{aligned}{\frac {\pi }{2}}&=\prod _{n=1}^{\infty }{\frac {4n^{2}}{4n^{2}-1}}=\prod _{n=1}^{\infty }\left({\frac {2n}{2n-1}}\cdot {\frac {2n}{2n+1}}\right)\\[6pt]&={\Big (}{\frac {2}{1}}\cdot {\frac {2}{3}}{\Big )}\cdot {\Big (}{\frac {4}{3}}\cdot {\frac {4}{5}}{\Big )}\cdot {\Big (}{\frac {6}{5}}\cdot {\frac {6}{7}}{\Big )}\cdot {\Big (}{\frac {8}{7}}\cdot {\frac {8}{9}}{\Big )}\cdot \;\cdots \\\end{aligned}}}
.
![{\displaystyle {\begin{aligned}{\frac {\pi }{2}}&=\prod _{n=1}^{\infty }{\frac {4n^{2}}{4n^{2}-1}}=\prod _{n=1}^{\infty }\left({\frac {2n}{2n-1}}\cdot {\frac {2n}{2n+1}}\right)\\[6pt]&={\Big (}{\frac {2}{1}}\cdot {\frac {2}{3}}{\Big )}\cdot {\Big (}{\frac {4}{3}}\cdot {\frac {4}{5}}{\Big )}\cdot {\Big (}{\frac {6}{5}}\cdot {\frac {6}{7}}{\Big )}\cdot {\Big (}{\frac {8}{7}}\cdot {\frac {8}{9}}{\Big )}\cdot \;\cdots \\\end{aligned}}}](./_assets_/df59bf8aa67b6dff8be6cffb4f59777cea828454.svg)
.