10 - MATH 138 Winter 2011 Solution Note that Assignment 10...

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MATH 138 Winter 2011 Assignment 10 Topics: Representation of a function as a power series, Taylor and Maclaurin series. Due: 11 a.m. Friday, March 25. 1. Find a power series representation for the functions and determine the radius of con- vergence. (a) ln( x 2 + 4) Solution: Recall that ln(1 - x ) = - R 1 1 - x dx = - R ∑ n 0 x n dx = C - n 0 x n +1 ( n +1) = C - n 1 x n n for | x | < 1 . Evaluating at x = 0 , we see that C = 0 . Since ln( x 2 + 4) = ln ± 4 ( 1 + x 2 4 ) ² = ln(4) + ln ( 1 + x 2 4 ) , we sub- stitute - x 2 4 in the previous series. So we have ln( x 2 + 4) = ln(4) - X n 0 ( - x 2 4 ) n n = ln(4) + X n 0 ( - 1) n +1 x 2 n n 4 n for | x 2 / 4 | < 1 , that is for | x | < 2 . Hence the radius of convergence is 2 . (b) x 2 ( a 3 - x 3 ) 2 Solution: Note that x 2 ( a 3 - x 3 ) 2 = 1 3 d dx ( 1 a 3 - x 3 ) = 1 3 a 3 d dx ( 1 1 - x 3 a 3 ) = 1 3 a 3 d dx ( X n 0 x 3 n a 3 n ) = 1 3 a 3 d dx X n 0 3 nx 3 n - 1 a 3 n = d dx X n 1 nx 3 n - 1 a 3( n +1) . Since we have used the geometric se-
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10 - MATH 138 Winter 2011 Solution Note that Assignment 10...

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