Math 2414 Section 9 4 Working with Taylor Series x 2 2 cos x 2 lim 3x 4 Evaluate x 0 5 1 lim 6 x sin 6 x 4 x 2 x x Evaluate Differentiate the Maclaurin series for f x sin x y t y 2 subject to the initial Find a power series solution of the differential equation y 0 6 condition Identity the function represented by the power series When values of x are substituted into a convergent power series the result may be a series representation of a familiar real number The following example illustrates some techniques Use the Maclaurin series for f x tan 1 x to evaluate 1 1 1 K 3 5 Let f x and ex 1 1 x for x 0 and f 0 1 Use the Maclaurin series for f to evaluate f k k 1 k 1 Identify the function represented by the power series convergence 1 2x k 0 k k and give its interval of Determine the interval of convergence of the power series represents on this interval 1 k 0 4 k k k x2k and find the function it
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