Chapter4

Chapter4 - Chapter 4: Sequences and Mathematical Induction...

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Chapter 4: Sequences and Mathematical Induction February 13, 2008
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Outline 1 4.1 Sequences 2 4.2 Mathematical Induction I 3 4.3 Mathematical Induction II 4 4.4 Strong Mathematical Induction
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Sequences Sequence: an ordered set of elements (most often, numbers) a m , a m + 1 , a m + 2 , . . . , a n The elements of a sequence are called terms of the sequence. a i is called the i -th term of the sequence. Sequences are often infinite: a m , a m + 1 , a m + 2 , . . .
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Sequences can be given in two ways: 1 analytically , when each term a k is given by some known formula involving k : a k = f ( k ) where f is a function. 2 recursively , if the first m terms are given explicitly (these are called initial conditions ) and the rest of the terms a m + 1 , a m + 2 , . . . are given by a recursive formula a n = f ( a n - 1 , a n - 2 , . . . , a i - m ) , n > m
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Examples (a) a k = ( - 1 ) k 2 k - 1 2 , 1 4 , - 1 8 , . . . (b) a 1 = 1 and a k = k · a k - 1 . This is an example of a recursively defined sequence. a 2 = 2 · 1 = 2 a 3 = 3 · a 2 = 6 a 4 = 4 · a 3 = 24 . . . In fact, this sequence is a k = k !
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Series Definition A series is the sum of all terms of a sequence. Given a sequence a i and two positive integers m and n , such that n m n X i = m a i = a m + a m + 1 + . . . + a n If the sequence is infinite, we write X i = m a i = a m + a m + 1 + a m + 2 + . . . [Of course, with infinite series, this sum may not exist as a real number; i.e. such a series may not converge.]
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Examples (a) 6 X i = 2 i = 2 + 3 + 4 + 5 + 6 = 20 (b) 3 X i = 3 i = 3 (c) 5 X i = 1 ( - 1 ) i = - 1 + 1 - 1 + 1 - 1 = - 1 (d) 4 X i = 1 ( - 1 ) i i + 1 = - 1 2 + 1 3 - 1 4 + 1 5 = - 13 60
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Change of Variable Let a i be a sequence and consider the two series n X i = 1 a i = a 1 + a 2 + . . . + a n - 1 + a n n + 1 X i = 2 a i - 1 = a 1 + a 2 + . . . + a n - 1 + a n Therefore, these two are the same series. In general, given a series n i = m a i , we can change the variable by setting j = i + k ( k Z ) to get n + k X j = m + k a j - k .
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Example Transform the sum n - 1 X i = 1 i ( n - i ) 2 by making the change of the variable j = i - 1. Solution: Since j = i - 1, we have i = j + 1, so: n - 2 X j = 0 j + 1 ( n - ( j + 1 )) 2 = n - 2 X j = 0 j + 1 ( n - j - 1 ) 2
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Product Notation Given a sequence a i , we define the product of the sequence as n Y i = m a i = a m · a m + 1 · . . . · a n Example 4 Y k = 1 k k + 1 = 1 2 · 2 3 · 3 4 · 4 5 = 1 5
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Definition For each positive number n , n factorial denoted n ! is defined as the product of all integers from 1 to n n ! = n Y i = 1 i = n · ( n - 1 ) · . . . · 2 · 1 We also define 0 ! = 1 For example, 7 ! = 7 · 6 · 5 · 4 · 3 · 2 · 1 = 5040
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Theorem If a i and b i are two sequences and n m: 1 n X k = m a k + n X k = m b k = n X k = m ( a k + b k ) 2 c · n X k = m a k = n X k = m c · a k 3 ± n Y k = m a k ! · ± n
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This note was uploaded on 04/17/2008 for the course MTH 314 taught by Professor Forgot during the Spring '08 term at Ryerson.

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Chapter4 - Chapter 4: Sequences and Mathematical Induction...

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