3_induction_Stirling_s_Formula

3_induction_Stirling_s_Formula - Discrete Mathematics Dr....

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Unformatted text preview: Discrete Mathematics Dr. Fred Phelps Lecture 3 Induction and Stirlings Formula LVP 2.1 Mathematical Induction A tool to prove formulae for every What is the sum of the first n odd numbers, i.e. Lets experiment: . n 1 (2 1)? n i i =- Conjecture: Proof: We will use induction . Step 1: Prove the statement holds for n =1. (Like getting on the first rung of a ladder.) Step 2: Assume the statement is true for n = k. (Or you can assume its true for all integers 1, 2, k . Like reaching the k th step of a 2 1 (2 1) . n i i n =-= Step 3: Using the induction assumption (step 2) prove that the statement holds for n=k+1 . (If we can get to step k on the ladder then we can get to step k+1 one more step). Stairway to Heaven So how high can we climb if we can get to the first step and we can always climb one more step? There is no limit! 2 1 (2 1) . n i i n =-= Step 1: Step 2: Assume that Step 3: Need to prove that Split off the last term of the sum: which is what we needed to prove. Proof that 1 2 1 (2 1) 1 1 . i i =-= = 2 1 (2 1) . k i i k =-= 1 2 1 (2 1) ( 1) . k i i k + =-= + 1 2 2 1 1 (2 1) (2 1) (2 1) 2 1 ( 1) . k k i i i i k k k k + = =-=-+ + = + + = + Alternate Definition The condition (WO) is called the Well-ordering principle. So the principle of induction follows from logic and the properties of the natural numbers. In the strong form, in order to get to the n+1 st step of the ladder, we use not only the n th step but all the steps below, which we know we can reach. LPV 2.1 Subset counting LPV 2....
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This note was uploaded on 10/15/2011 for the course MATHS 100 taught by Professor Fredphelps during the Spring '11 term at Jordan University of Science & Tech.

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3_induction_Stirling_s_Formula - Discrete Mathematics Dr....

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