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M257-316Notes_Lecture2

# M257-316Notes_Lecture2 - Chapter 2 Lecture 2 Preliminaries...

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Chapter 2 Lecture 2 - Preliminaries 2.1 Sequences and Series of Numbers: A Sequence of Numbers: 1 , 1 2 , 1 3 ,..., 1 n ,... 1 , 1 2 , 1 4 ( 1 2 ) n 1 Notation { a n } a n = 1 n { b n } b n = ( 1 2 ) n 1 A Series of Numbers: 1+ 1 2 + 1 3 + ··· + 1 n + = X n =0 1 n X r =0 a n (2.1) Does this inFnite sum yield a Fnite result? µ 1 2 + µ 1 2 2 + + µ 1 2 n 1 = X n =1 µ 1 2 n 1 X n =1 b n (2.2) Note: In order to sum to a Fnite number the terms of the sequence must tend to 0 as n →∞ . Divergence Test: lim a n 6 =0 X 0 a n diverges. (2.3) EG: a n =1 1+1+ +1+ ···→∞ . Integral Test: Does n =1 1 n converge? 11

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Lecture 2 - Preliminaries Consider Z 1 dx x < 1+ 1 2 + 1 3 + ··· + 1 n + = X n =1 1 n (2.4) Now Z 1 dx x = lim T →∞ T Z 1 dx x = lim T →∞ (ln T ln 1) = But Z 1 dx x < X n =1 1 n Therefore X n =1 1 n = . Example: For what values of p will the series X n =1 1 n p (2.5) converge? We consider the integral: R 1 1 x p dx = lim T →∞ T R 1 dx x p = ± x 1 p 1 p R T 1 p 6 =1 ln x R T 1 p = ½ 1 p 1 p>
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M257-316Notes_Lecture2 - Chapter 2 Lecture 2 Preliminaries...

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