binomial-coeffs

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

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Unformatted text preview: Binomial Coefficients Jonathan L.F. King University of Florida, Gainesville FL 32611-2082, USA [email protected] 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 !...
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