hw3solns - CMPS 101 Summer 2009 Homework Assignment 3...

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1 CMPS 101 Summer 2009 Homework Assignment 3 Solutions 1. (1 Point) The last exercise in the handout entitled Some Common Functions . Use Stirling's formula to prove that Θ = n n n n 4 2 . Proof: By Stirling’s formula ( ) ( ) 2 2 2 ) / 1 ( 1 2 ) 2 / 1 ( 1 2 2 2 ) ! ( )! 2 ( )! 2 ( ! )! 2 ( 2 Θ + Θ + = = - = n e n n n e n n n n n n n n n n n n π ( ) ( ) 2 2 2 ) / 1 ( 1 ) 2 / 1 ( 1 4 1 ) / 1 ( 1 ) 2 / 1 ( 1 2 n n n n n n n n Θ + Θ + = Θ + Θ + = so that ( ) 1 ) / 1 ( 1 ) 2 / 1 ( 1 1 4 2 2 Θ + Θ + = n n n n n n as n The result now follows since < < 1 0 . /// 2. (2 Points) (Exercise 1 from the induction handout) Prove that for all 1 n : 2 1 3 2 ) 1 ( + = = n n i n i . Do this twice: a. (1 Point) using form I Ia of the induction step. b. (1 Point) using form IIb of the induction step. Proof: Let ) ( n P be the equation 2 1 3 2 ) 1 ( + = = n n i n i . I.
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hw3solns - CMPS 101 Summer 2009 Homework Assignment 3...

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