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Would you like this assignment to be marked? YES NO MA104 Lab Report 10 - Power Series, Taylor and Maclaurin Series Name: Student Number: Fall 2018 1. [8 marks ] Find the centre of convergence, the radius of convergence, and the interval of convergence of the series n =1 4 n n x n . (Remember to check convergence at endpoints!) Ratio Test: lim n →∞ a n +1 a n = lim n →∞ 4 n +1 x n +1 n + 1 · n 4 n x n = lim n →∞ 4 x r n n + 1 = lim n →∞ 4 x r 1 1 + 1 /n - 1 < 4 x < 1 = | 4 x | ⇒ - 1 4 < x < 1 4 Checking Endpoints: At x = 1 4 : n =1 4 n n 1 4 n = n =1 1 n which is a divergent p-series ( p = 1 2 < 1). At x = - 1 4 : n =1 4 n n - 1 4 n = n =1 ( - 1) n n which converges by the Alternating Series Test: Let b n = 1 n i) b 0 ( n ) = - 1 2 n 3 / 2 < 0 for all n 1 ii) lim n →∞ b n = lim n →∞ 1 n = 0 The interval of convergence is - 1 4 , 1 4 with a radius of convergence R = 1 4 - - 1 4 2 = 1 4 . 2. [6 marks ] Consider f ( x ) = 7 4 - 3 x . (a) Use the fact that 1 1 - x = n =0 x n for | x | < 1 to show that f ( x ) = n =0 7 4 3 x 4 n . f ( x ) = 7 4 - 3 x = 7 4 · 1 1 - 3 x 4 = n =0 7 4 3 x 4 n (b) What is the interval of convergence for n =0 7 4 3 x 4 n ?

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