20d_04s_1e

20d_04s_1e - or conditionally convergent You must give...

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Math 20D First Midterm 48 points April 21, 2004 Print Name, ID number and Section on your blue book. BOOKS and CALCULATORS are NOT allowed. One side of one page of NOTES is allowed. You must show your work to receive credit. 1. (5 points) Let a n x n be the Maclaurin series for f ( x ) = (cos x ) sin( x 2 ). Compute a 1 , a 2 , a 3 a 4 and a 5 . 2. (15 points) Find the radii of convergence of the following power series. You must give correct reasons for your answers to receive credit. (a) s n =0 x n (2 n )! (b) s n =0 (2 x - 3) n (c) s n =0 n 3 x n +1 n 2 + 1 3. (20 points) Determine if each of the following series is convergent or divergent. If the series is convergent and the terms alternate in sign, determine the series is absolutely
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Unformatted text preview: or conditionally convergent. You must give correct reasons for your answers to receive credit. (a) ∞ s n =1 (-1) n 3 2 n n 3 2 3 n (b) ∞ s n =1 e-√ n √ n (c) ∞ s n =1 (-1) n sin(1 /n ) (d) ∞ s n =1 ( n !) 2 (2 n )! 4. (8 points) Let ∑ a n x n be the Maclaurin series for f ( x ) and let ∑ b n x n be the Maclau-rin series for f (-x ). (a) Express b n in terms of a n . (b) Suppose f ( x ) is an even function; that is, f (-x ) = f ( x ). Show that, whenever n is odd, a n = 0. END OF EXAM...
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