A9 - =1 n 2 2 n = 6 by evaluating this power series at x =...

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Math 128: Assignment 9 due Wednesday, Dec 7 by 4:30 pm Koray Karabina ([email protected]) 1. In parts (a)–(e), determine whether the series converge absolutely, converge conditionally, or diverge. (a) X n =2 ( - 1) n 1 n ln n (b) X n =1 ( - 1) 2 n 2 n (c) X n =1 ( - 1) n n (d) X n =1 ( - 1) n +1 4 n (e) X n =1 n ! 100 n 2. In parts (a)–(d), find the radius of convergence and interval of convergence of the series (do not forget to check the end points). (a) X n =1 ( - 2) n x n 4 n (b) X n =0 x 2 n n + 1 (c) X n =1 1 n x + 2 2 n (d) X n =0 (1 + 5 n ) n ! x n 3. In this question, you will (eventually) prove that X n =1 n 2 2 n = 6. (a) Using the power series representation of 1 / (1 - x ) 2 = n =1 nx n - 1 (which was derived in one of our lectures), find a power series representation for x/ (1 - x ) 2 . (b) Using part (a), find a power series representation for (1 + x ) / (1 - x ) 3 . Hint: Differentiation is useful. (c) Using part (b), find a power series representation for x (1 + x ) / (1 - x ) 3 . Conclude that
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Unformatted text preview: =1 n 2 2 n = 6 by evaluating this power series at x = 1 / 2. 4. Determine the interval of convergence and the sum of the series: 1-4 x + 16 x 2-64 x 3 + ··· = ∞ X n =0 (-1) n (4 x ) n Hint: Use ∑ ∞ n =0 x n = 1 / (1-x ) for-1 < x < 1. 5. In parts (a)–(c), find a power series representation for the function and determine the radius of convergence. (a) ln(7-x ) (b) x 3 ( x-2) 2 (c) arctan ( x/ 3) 6. Find Maclaurin series (Taylor series centered at 0) for the functions in parts (a)–(c). (a) sin 2 ( x ) (b) cosh x (c) e-x 2 / 3 Hints: Part (a): sin 2 ( x ) = (1-cos 2 x ) / 2, Part (b): Recall that cosh x = ( e x + e-x ) / 2. 7. In parts (a)–(b), use series to evaluate the limit. (a) lim x → x-arctan x x 3 (b) lim x → 1-cos x 1 + x-e x...
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