2012A12 - + x + x 2 + x 3 + ... In other words, 1 1-x = 1 +...

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Problem Set 12 - Due by Thursday, March 22, 2012 Posted/Revised: March 14, 2012 Note: SHE = Calculus (tenth edition) by Salas, Hille, and Etgen Problems marked with * are to be handed in during lecture or in SS1071. Please write your three character tutorial code at the top of you assignment (e.g., M2A, T3B, etc.) 1. SHE 11.3 : 1, 3, 5, 7, 9, 17, 29, 31, 33, 35, 37, 39, 49, 55, 57, 59, 61, 63, 65 SHE 11.4 : 1, 3, 5, 7, 9, 11, 15, 21, 27, 37, 43 SHE 12.2 : 3, 7, 9, 11, 13, 15, 17, 21, 29, 33 SHE 12.3 : 1, 3, 5, 9, 11, 13, 21, 27, 29, 33, 35, 37, 47, 49 SHE 12.4 : 1, 3, 5, 7, 9, 17, 19, 23, 29, 31, 37, 43, 45 All recommended problems (including the above) are listed at www.math.toronto.edu/lshorser/MAT137/MAT137RecommendedProblems.pdf 2. (a) SHE 11.3 # * 52 (b) SHE 11.4 # * 48 (c) SHE 12.2 # * 22 (d) SHE 12.3 # * 16 , * 28 (e) SHE 12.4 # * 14 , * 16 , * 44 3. When ± x ± < 1, the series expansion for 1 1 - x is 1
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Unformatted text preview: + x + x 2 + x 3 + ... In other words, 1 1-x = 1 + x + x 2 + x 3 + ... * (a) Us the above series to nd a series expansion for f ( x ) = x 1 + x 2 when x &lt; 1. * (b) By integrating each term in your answer to (a), nd a series expansion for 1 2 ln ( 1 + x 2 ) , when x &lt; 1. * 4. The power series expansion for the exponential function is given by: e x = 1 + x + x 2 2! + x 3 3! + ... 1 Let P n ( x ) = 1 + x + x 2 2! + ... + x n n ! In other words, P n ( x ) is the sum of the rst n + 1 terms in the series. * (a) Which is closer to the value of e 7 : P 5 ( 7 ) or P 9 ( 7 ) ? Explain your answer. * (b) e x can be written as k = x k k ! . Use the ratio test to see if this series converges when x = 2 3 and when x = 6. * (c) Will k = x k k ! converge for all real values of x ? Why or why not? 2...
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2012A12 - + x + x 2 + x 3 + ... In other words, 1 1-x = 1 +...

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