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Unformatted text preview: Mathematics 222 Calculus III Assignment 2 1. For the following power series, find (a) the radius of convergence (b) the interval of convergence, discussing the endpoint convergence when the radius of convergence is finite. (i) (ii) (iv) 2. Given f (x) =
1 (x-1)n 1 3n n , 1
2 1 n n 1- xn , (v) J0 (x) = (iii) (-1)n 0 (n!)2 4n xn [log(n+1)]n , (-1)n (x+1)2n 4n n2 log(n) x n 2
2 (-1)n+1 (x - 5)n n5n find the interval of convergence of the Taylor series expansions around x = 5 of the following (a) f (x), 3. If f (x) =
x 1-e-t dt t 0 (b) f (x), (c) x (f (t)dt 5 (a) find a power series for f (x) about x = 0 (b) find the interval of convergence of this series. (c) compute f (0.4) to four decimal place accuracy justifying your answer 4. (a) Obtain the Taylor series for f (x) = (c) use the series to compute f (6) (1) 5. find the first three non-zero terms of the Maclaurin expansion of y = f (x) where the function is defined implicitly by x2 + xy + y 2 = 1. Also estimate the error approximating f(0.1) using the first two nonzero terms of this series.
3 x2 -x-2 about x = 1. (b) find the interval of convergence of this series. ...
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