binomial-coeffs - Binomial Coefficients Jonathan L.F. King...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Binomial Coefficients Jonathan L.F. King University of Florida, Gainesville FL 32611-2082, USA squash@math.ufl.edu Webpage http://www.math.ufl.edu/ squash/ 11 October, 2006 (at 07:17 ) Entrance. Use ineq. for inequality. 1 : Theorem. Consider inequality ( 2 n n ) 1 * > 1 2 n 4 n , and these two: B p 1 2 + n 4 n 3 * > 2 n n ! 2 * > A n 4 n . Setting B := 1 and A := 1 2 , inequalities (3 * ,2 * ) hold for all posints n , as does (1 * ) . Proof. Let b n denote some unknown positive function of n , and let n := 1 / b n . Each of the three bounds above is OTForm b n 4 n . Thus each inequality has the form 2 n n ! 4 n b n, ( n ) : where relation is either > or 6 . Assume that we have verified ( ) for some base-case value of n . The induction step is ( n- 1) = ( n ). Here is the algebra: 2 n n ! = 2 n- 1 n- 1 ! [2 n- 1] 2 n n n = 4 2 n- 1 n- 1 !...
View Full Document

Ask a homework question - tutors are online