Wallis formula
A formula which expresses the number $ \pi /2 $ as an 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 [1] in his calculation of the area of a disc; it is one of the earliest examples of an infinite product.
References
| [1] | J. Wallis, "Arithmetica infinitorum" , Oxford (1656) |
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
| [a1] | T.M. Apostol, "Calculus" , 2 , Blaisdell (1964) |
| [a2] | C.H. Edwards jr., "The historical development of the calculus" , Springer (1979) |
| [a3] | P. Lax, S. Burstein, A. Lax, "Calculus with applications and computing" , 1 , Springer (1976) |
| [a4] | D.J. Struik (ed.) , A source book in mathematics: 1200–1800 , Harvard Univ. Press (1986) |
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