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© 1999-2010 by Francis Edward Su

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!

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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