Na Galipedia, a Wikipedia en galego.
Comparación da converxencia do produto de Wallis (asteriscos en rubio) e diversas series infinitas históricas para π. Sn é a aproximación despois de tomar n termos. Cada subsequente gráfica magnifica a área sombreada horizontalmente por 10 veces. (clicar para detalle) O produto de Wallis é a representación de π produto infinito :
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}}}
Publicado por John Wallis en 1656.[ 1]
O produto de Wallis é, retrospectivamente, un fácil corolario do que foi posterior produto infinito de Euler para a función seno .
sin
x
x
=
∏
n
=
1
∞
(
1
−
x
2
n
2
π
2
)
{\displaystyle {\frac {\sin x}{x}}=\prod _{n=1}^{\infty }\left(1-{\frac {x^{2}}{n^{2}\pi ^{2}}}\right)}
Sexa
x
=
π
2
{\displaystyle x={\frac {\pi }{2}}}
⇒
2
π
=
∏
n
=
1
∞
(
1
−
1
4
n
2
)
⇒
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
⋯
{\displaystyle {\begin{aligned}\Rightarrow {\frac {2}{\pi }}&=\prod _{n=1}^{\infty }\left(1-{\frac {1}{4n^{2}}}\right)\\[6pt]\Rightarrow {\frac {\pi }{2}}&=\prod _{n=1}^{\infty }\left({\frac {4n^{2}}{4n^{2}-1}}\right)\\[6pt]&=\prod _{n=1}^{\infty }\left({\frac {2n}{2n-1}}\cdot {\frac {2n}{2n+1}}\right)={\frac {2}{1}}\cdot {\frac {2}{3}}\cdot {\frac {4}{3}}\cdot {\frac {4}{5}}\cdot {\frac {6}{5}}\cdot {\frac {6}{7}}\cdots \end{aligned}}}
[ 1]
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/df59bf8aa67b6dff8be6cffb4f59777cea828454)
![{\displaystyle {\begin{aligned}\Rightarrow {\frac {2}{\pi }}&=\prod _{n=1}^{\infty }\left(1-{\frac {1}{4n^{2}}}\right)\\[6pt]\Rightarrow {\frac {\pi }{2}}&=\prod _{n=1}^{\infty }\left({\frac {4n^{2}}{4n^{2}-1}}\right)\\[6pt]&=\prod _{n=1}^{\infty }\left({\frac {2n}{2n-1}}\cdot {\frac {2n}{2n+1}}\right)={\frac {2}{1}}\cdot {\frac {2}{3}}\cdot {\frac {4}{3}}\cdot {\frac {4}{5}}\cdot {\frac {6}{5}}\cdot {\frac {6}{7}}\cdots \end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/74ce7488d92c2708916455e66c7770dbfed21150)
