note14 - Math021, week 14 Definition 14.1 A series ∑ ∞...

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Unformatted text preview: Math021, week 14 Definition 14.1 A series ∑ ∞ n =1 a n converges absolutely if ∑ ∞ n =1 | a n | converges. Theorem 14.2 A series converges if it converges absolutely. proof: Let ∑ ∞ n =1 a n be a series which converges absolutly. Define a + n = max { a n , } , and a- n = max {- a n , } for all n . Then, 0 ≤ a + n ≤ | a n | for all n and since ∑ ∞ n =0 | a n | converges by hypothesis, we conclude that ∑ ∞ n =1 a + n converges by comparison test. Similarly, ∑ ∞ n =1 a- n converges. Therefore, ∞ X n =1 a n = ∞ X n =1 ( a + n- a- n ) converges also. Example 14.3 Determine if the series ∞ X n =1 (- 1) n n 3 / 2 converges. solution: Since ∞ X n =1 | (- 1) n n 3 / 2 | = ∞ X n =1 1 n 3 / 2 converges from a certain previous example, ∞ X n =1 (- 1) n n 3 / 2 converges absolutely. Hence, it converges. Power Series 1 Definition 14.4 A power series is a function f defined by f ( x ) = ∞ X n =0 c n ( x- a ) n for certain numbers a , c , c 1 ,.... We also say that such a power series is centered at a . Example 14.5 The function f defined by f ( x ) = ∞ X n =0 x n is a power series centered at . It is a geometric series with common ratio x . Hence, f ( x ) = 1 1- x when | x | < 1 . We also know that the series defining f diverges when | x | > 1 . Therefore, f is defined over (- 1 , 1) . Theorem 14.6 Let f ( x ) = ∞ X n =0 c n ( x- a ) n be a power series centered at a . There is a number R ≥ , called the radius of convergence of the given series, so that the series converges absolutely when | x- a | < R , and the series diverges when...
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This note was uploaded on 02/13/2012 for the course MATH 021 taught by Professor Luxnu during the Fall '10 term at HKUST.

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note14 - Math021, week 14 Definition 14.1 A series ∑ ∞...

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