153-t3sol - Math 153 - Spring 2010 NAME Signature Test 3 5...

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Math 153 - Spring 2010 NAME Signature Test 3 5 May 2010 Answer the following questions. The answers must be clear, intelligible, and you must show your work. Provide explanation for all your steps. Your grade will be determined by adherence to these criteria. Use of books, notes and calculators is strictly forbidden. The point value of each problem is given in the left hand margin. (6 pts.) 1. Use power series to evaluate the following limit: lim x ! 0 sin x ± x x 3 lim x ! 0 sin x ± x x 3 = lim x ! 0 1 x 3 1 X n =0 ( ± 1) n x 2 n +1 (2 n + 1)! ± x ! = lim x ! 0 1 x 3 ± ± x 3 3! + x 5 5! ± x 7 7! + ²²² ² = lim x ! 0 ± ± 1 3! + x 2 5! ± x 4 7! + ²²² ² = ± 1 6 1
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(10 pts.) 2. Find a power series representation for the function and determine the radius of convergence. f ( x ) = ln(1 + x 2 ) We ±rst compute f 0 ( x ) = 2 x 1 + x 2 and we ±nd a power series representation for the derivative: f 0 ( x ) = 2 x 1 ± ( ± x 2 ) = 2 x 1 X n =0 ( ± x 2 ) n = 2 x 1 X n =0 ( ± 1) n x 2 n = 2 1 X n =0 ( ± 1) n x 2 n +1 Integrate the above relation to get: f ( x ) = 2 Z 1 X n =0 ( ± 1) n x 2 n +1 dx = C + 2 1 X n =0 ( ± 1) n x 2
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This note was uploaded on 04/05/2011 for the course MATH 153 taught by Professor Rempe during the Spring '08 term at Ohio State.

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153-t3sol - Math 153 - Spring 2010 NAME Signature Test 3 5...

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