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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 < 1. * (b) By integrating each term in your answer to (a), nd a series expansion for 1 2 ln ( 1 + x 2 ) , when x < 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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- Spring '08