132FinalReview.pdf - Lecture 21 Final Review 1 Radius and...

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Lecture 21: Final Review. 1. Radius and Interval of convergence of power series. Power series n =0 c n ( x - a ) n . Following are steps to find radius and interval of convergence: 1. Let a n = c n ( x - a ) n , compute the absolute ratio | a n +1 a n | . 2. Compute the lim n →∞ | a n +1 a n | = L . 3. Set L < 1 and write it into the form | x - a | < R . So at this step a, R are known. 4. Plug in x = a - R, a + R respectively, check whether the power series converges at these endpoints of the open interval ( a - R, a + R ). 2. Substitution, differentiation and integration to find power series repre- sentation. Remember two basic functions and their power series: 1 1 - x = X n =0 x n . and 1 (1 - x ) 2 = X n =0 nx n - 1 , or X n =1 nx n - 1 . Both have interval of convergence | x | < 1. Two type of rational functions: 1. If a rational function is like q ( x ) a + bx r , then write q ( x ) a + bx r = q ( x ) · 1 a (1 - ( - bx r a )) = q ( x ) a · 1 (1 - ( - bx r a )) , then substitute - bx r a for x in the power series n =0 x n . 2. If a rational function is like q ( x ) ( a + bx r ) 2 , then write q ( x ) ( a + bx r ) 2 = q ( x ) · 1 a 2 (1 - ( - bx r a )) 2 = q ( x ) a 2 · 1 (1 - ( - bx r a )) 2 , then substitute - bx r a for x in the power series n =0 nx n - 1 .
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