View source for Wallis formula

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A formula which expresses the number  $  \pi /2 $
as an [[Infinite product|infinite product]]:

$$ \tag{1 }
{
\frac \pi {2}
 }  = \ 
\left ( {
\frac{2}{1}
 } \cdot {
\frac{2}{3}
 } \right ) \left ( {
\frac{4}{3}
 } \cdot
{
\frac{4}{5}
 } \right ) \dots \left ( {
\frac{2k}{2k-1}
 } \cdot
{
\frac{2k}{2k+1}
 } \right ) \dots =
$$

$$ 
= \ 
\lim\limits _ {m \rightarrow \infty }  \prod _ { k=1 } ^ { m }  
\frac{( 2k)  ^ {2} }{( 2k- 1)( 2k+ 1) }
 .
$$

There exist other variants of this formula, e.g.:

$$ \tag{2 }
\sqrt \pi  = \ 
\lim\limits _ {m \rightarrow \infty } \ 

\frac{( m!)  ^ {2} \cdot 2  ^ {2m} }{( 2m)!  \sqrt m }
 .
$$

Formula (1) was first employed by J. Wallis [[#References|[1]]] in his calculation of the area of a disc; it is one of the earliest examples of an infinite product.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J. Wallis,   "Arithmetica infinitorum" , Oxford  (1656)</TD></TR></table>

====Comments====
Formula (1) is a direct consequence of Euler's product formula

$$ 
\sin  z  =  z \prod _ { n=1 } ^  \infty   \left ( 1 - \frac{z  ^ {2} }{n  ^ {2} \pi  ^ {2} } \right )
$$
with  $z = \pi /2 $.

It can also be obtained by expressing  $  \int _ {0} ^ {\pi /2 } \sin  ^ {2m}  x  dx $
and  $  \int _ {0} ^ {\pi /2 } \sin  ^ {2m+1}  x  dx $ in terms of  $  m $, and by showing that

$$ 

\frac{\int\limits _ { 0 } ^ {  \pi   /2 } \sin  ^ {2m}  x  dx }{\int\limits _ { 0 } ^ {  \pi   /2 } \sin  ^ {2m+1}  x  dx }
  \rightarrow  1 \  ( m\rightarrow \infty ).
$$

Formula (2) can be derived from (1) by multiplying the numerator and the denominator of  $  \prod _ {k=1}  ^ {m} ( 2k)  ^ {2} / ( 2k- 1)( 2k+ 1) $
by  $  2  ^ {2} \cdot 4  ^ {2} \dots ( 2m)  ^ {2} $.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  T.M. Apostol,   "Calculus" , '''2''' , Blaisdell  (1964)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  C.H. Edwards jr.,   "The historical development of the calculus" , Springer  (1979)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  P. Lax,   S. Burstein,   A. Lax,   "Calculus with applications and computing" , '''1''' , Springer  (1976)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  D.J. Struik (ed.) , ''A source book in mathematics: 1200–1800'' , Harvard Univ. Press  (1986)</TD></TR></table>