Sponsored by the Rutgers University Department of Mathematics and the

Center for Discrete Mathematics and Theoretical Computer Science (DIMACS)

**Co-organizers:****Andrew Baxter**, Rutgers University, baxter{at} math [dot] rutgers [dot] edu**Doron Zeilberger**, Rutgers University, zeilberg {at} math [dot] rutgers [dot] edu

Title: Unexpected connections between three famous old formulas for π

Speaker: **Tom Osler**, Rowan University

Date: Thursday, November 11, 2010 5:00pm

Location: Hill Center, Room 705, Rutgers University, Busch Campus, Piscataway, NJ

Abstract:

In 1593 Vieta produced an infinite product for 2/π in which the factors are nested radicals. In 1656 John Wallis published his "Arithmetica Infinitorum", in which he gave another infinite product for π/2. The Wallis product is very different as the factors are rational numbers. In the same book, Wallis published a continued fraction for 4/π 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?