# A find a power series representation of the function

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(a) Find a power series representation of the function f ( x ) = 1 1 + x 2 , and state the interval on which the series converges to the function. (b) Use the series in (a) to find a power series representation of the function g ( x ) = x 5 1 + x 2 .

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CALCULUS II, TEST IV 7 Problem 5 Answer all the following questions. (a) Find the Maclaurin series representation of the function f ( x ) = sin( x 2 ). (Hint: The Maclaurin series of sin( x ) might prove useful here, if need be!) (b) Use the series in (a) to evaluate the integral Z sin( x 2 ) dx as a power series. (c) Use the series in (b) to write out the Maclaurin series representation of Z 1 0 sin( x 2 ) dx (Do not compute and add the terms of your series!) (d) Find the minimum number of terms you need in the series in (c) to approximate Z 1 0 sin( x 2 ) dx with an error less than 10 - 3 = 0 . 001? (Show your work!)
CALCULUS II, TEST IV 8 SCRATCH PAPER (Scratch paper will not be graded!)

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CALCULUS II, TEST IV 9 SCRATCH PAPER (Scratch paper will not be graded!)
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