Math 102 (Summer session I, 2012) Homework 9 A/B/C on power series (and as representations of functions) Due date does not exist Things to keep in mind: Which theorem or test apply to the sequence or series in question? Not all techniques are applicable in all cases! HW9-A on power series 1 Find the radius convergence and the interval of convergence of the series 2 Find the ROC and the IOC of the series 3 Find the ROC and the IOC of the series ∞ √ X n=1 ∞ X (−1)n xn . n=1 n + 1 nxn . n n xn . n=1 ∞ X 10n xn . n3 n=1 ∞ X (x − 2)n 5 Find the ROC and the IOC of the series . nn n=1 4 Find the ROC and the IOC of the series ∞ X ∞ X (n!)k n x n=1 (kn)! 7 Let p and q be real numbers with p < q. Find four power series whose interval of convergence are (p, q), (p, q], [p, q), [p, q], respectively. 8 Is it possible to find a power series whose interval of convergence is [0, ∞)? Explain. 6 If k is a positive integer, find the radius of convergence of the series HW9-B on representations of functions as power series 3 and determine its IOC. 1 − x4 1 Find a power series representative of f (x) = and determine its IOC. x + 10 x Find a power series representative of f (x) = and determine its IOC. 9 + x2 x Find a power series representative of f (x) = 2 and determine its IOC. 2x + 1 1+x Find a power series representative of f (x) = and determine its IOC. 1−x (a) Find a power series representative for f (x) = ln(1 + x). (b) Use part (a) to find a power series for f (x) = x · ln(1 + x). (c) Use part (a) to find a power series for f (x) = ln(x2 + 1). Find a power series representation for f (x) = ln(5 − x) and determine the ROC. 1 Find a power series representative of f (x) = 2 3 4 5 6 7 HW9-C on Taylor and MacLaurin series 1 . (1 − x)2 Find the MacLaurin series for f (x) = ln(1 + x). Find the MacLaurin series for f (x) = cos(3x). Find the MacLaurin series for f (x) = x · ex . Find the Taylor series centered at a = 1 for f (x) = x4 − 3x2 + 1. Find the Taylor series centered at a = 3 for f (x) = ex . 1 Find the Taylor series centered at a = 9 for f (x) = √ . x 1 Find the Taylor series centered at a = 1 for 2 . Z 1x Use the Taylor inequalies to approximate x · cos(x3 )dx correct to three decimal places. Z0 0.4 √ Use the Taylor inequalies to approximate 1 + x4 dx correct to six decimal places. 1 Find the MacLaurin series for f (x) = 2 3 4 5 6 7 8 9 10 11 Use the Taylor inequalies to approximate Z00.5 0 2 x2 · e−x dx correct to four decimal places.