Unexpected connections between three famous old formulas for pi part 3





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Uploaded on Nov 18, 2010

Date: November 11, 2010
Speaker: Tom Osler, Rowan University
Title: Unexpected connections between three famous old formulas for pi
Abstract: In 1593 Vieta produced an infinite product for 2/pi in which the
factors are nested radicals. In 1656 John Wallis published his
"Arithmetica Infinitorum", in which he gave another infinite
product for pi/2. The Wallis product is very different as the
factors are rational numbers. In the same book, Wallis published
a continued fraction for 4/pi which he obtained from Lord
Brouncker. We will show how to morph the Wallis product into
Vieta's product. That this is possible is indeed a surprise. To
obtain this morphing we give a single formula that contains a
parameter "n". When n is zero, the formula produces the Wallis
product. When n = infinity, the formula gives Vieta's product.
As n increases 0, 1, 2, 3, ... we see the gradual transition form
a product of only rational numbers to a product of only nested
radicals. A second formula is given that allows us to morph
Brouncker's continued fraction into the Wallis product. Is there
a morphing between the product of Vieta and Brouncker's
continued fraction?

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