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© 1999-2010 by Francis Edward Su
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From the Fun Fact files, here is a Fun Fact at the Easy level:

Wallis' Formula

Here's a fun formula for Pi involving an infinite product, known as Wallis' Formula:


(Pi/2) = (2*2)
(1*3)
(4*4)
(3*5)
(6*6)
(5*7)
...

It is somewhat surprising that when you pull out every other pair of terms, you get a completely different kind of number!

Sqrt[2] = (2*2)
(1*3)
(6*6)
(5*7)
(10*10)
(9*11)
...

Presentation Suggestions:
You can also start with the infinite product, and ask if student can guess what it converges to, before you tell them the answer.

The Math Behind the Fact:
There is an infinite product formula for the sine function which yields Wallis' formula as a consequence. Infinite products are defined as the limit of the partial products, which are finite. This is similar to the way we define infinite sums!

How to Cite this Page:
Su, Francis E., et al. "Wallis' Formula." Math Fun Facts. <http://www.math.hmc.edu/funfacts>.

References:
    S. Vandervelde, Mathematics and Informatics Quarterly, 9 (1999), pp. 64-69.

Keywords:    pi formula
Subjects:    calculus, analysis, number theory
Level:    Easy
Fun Fact suggested by:   Sam Vandervelde
Suggestions? Use this form.
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