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Unformatted text preview: Math 311: Advanced Calculus Wolmer V. Vasconcelos Set 2 Spring 2008 Wolmer Vasconcelos (Set 2) Advanced Calculus Spring 2008 1 / 139 Outline 1 About Goals 2 Sequences 3 Limit Theorems 4 Monotone Sequences 5 Workshop #3 6 Series 7 BolzanoWeierstrass 8 Cauchy Criterion 9 Workshop #4 10 Typical EQuestions 11 Properties of Infinite Series 12 Current Workshop #5 Wolmer Vasconcelos (Set 2) Advanced Calculus Spring 2008 2 / 139 Some Goals Understand mathematical objects such as ∞ X n = a n = a + a 1 + a 2 + a 3 + ··· =? ∞ Y n = a n = a · a 1 · a 2 · a 3 + ··· =? The building blocks of these objects are a 1 , a 2 , a 3 ,..., a n ,...  {z } Wolmer Vasconcelos (Set 2) Advanced Calculus Spring 2008 3 / 139 Outline 1 About Goals 2 Sequences 3 Limit Theorems 4 Monotone Sequences 5 Workshop #3 6 Series 7 BolzanoWeierstrass 8 Cauchy Criterion 9 Workshop #4 10 Typical EQuestions 11 Properties of Infinite Series 12 Current Workshop #5 Wolmer Vasconcelos (Set 2) Advanced Calculus Spring 2008 4 / 139 Sequences of real numbers Definition A sequence is a function f whose domain is N . It can be represented as { f ( 1 ) , f ( 2 ) , f ( 3 ) ,... } { f ( ) , f ( 1 ) , f ( 2 ) , f ( 3 ) ,... } or { f ( n ) ,..., n ≥ n } We will first examine sequences of real numbers, f : N → R . Later we will study sequences of functions. Wolmer Vasconcelos (Set 2) Advanced Calculus Spring 2008 5 / 139 It allows us to look at real numbers in a concrete manner: If x = A . a 1 a 2 ··· a n ··· , where a i are the decimal digits, we form the sequence of rational numbers x = A x 1 = A . a 1 x 2 = A . a 1 a 2 x n = A . a 1 a 2 ··· a n , and so on Wolmer Vasconcelos (Set 2) Advanced Calculus Spring 2008 6 / 139 Examples We will look for features such as clustering 1 ( 1 , 1 2 , 2 3 , 3 4 ,... ) 2 ( c , c , c , c ,... ) 3 ( 1 , 1 2 , 2 3 , 3 4 ,... ) 4 ( 1 2 n ) ∞ n = 1 = ( 1 2 , 1 4 , 1 8 ,... ) 5 ( a n ) , a 1 = 1 , and a n + 1 = an + 1 2 6 ( a n ) , a n is the n th digit in the decimal expansion of π . 7 ( a n ) , a n = ( 1 + 1 / n ) n Wolmer Vasconcelos (Set 2) Advanced Calculus Spring 2008 7 / 139 Why Sequences? We use sequences to make sense of: ∑ n ≥ 1 a n : Series 1 + 1 / 2 2 + 1 / 3 2 + ··· + 1 / n 2 + ··· Question: How to handle ( a + a 1 + ··· + a n + ··· )( b + b 1 + ··· + b n + ··· ) ∑ m , n ≥ 1 a m , n : Double [multiple] Series X m , n 1 m 2 + n 2 Q n ≥ 1 a n : Infinite Products Y p ( 1 1 p ) , p prime number Wolmer Vasconcelos (Set 2) Advanced Calculus Spring 2008 8 / 139 Convergence of a Sequence Sequences are wonderful ways to represent data, but we are mostly interested is one of its aspects: Definition A sequence ( a n ) converges to a real number a if, for every positive real number , there exists an N ∈ N such that whenever n ≥ N it follows that  a n a  < ....
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 Spring '08
 vasconceles
 Calculus, Mathematical analysis, Limit of a sequence, Wolmer V. Vasconcelos

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