20d_04s_fs

# 20d_04s_fs - Math 20D Final Exam Solutions June 2004 1(a...

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Unformatted text preview: Math 20D Final Exam Solutions June 2004 1. (a) Since | sin x | ≤ 1, converges by comparison with 2 ∑ 1 /n 2 (the p test). (b) We have ∞ summationdisplay n =1 2 | sin n | n = sin 1 1 + ∞ summationdisplay n =2 | sin( n- 1) | n- 1 + | sin n | n ≥ ∞ summationdisplay n =2 | sin( n- 1) | + | sin n | n > 1 2 ∞ summationdisplay n =2 1 n , by | sin( x- 1) | + | sin x | > 1 / 2. Since (1 / 2) ∑ 1 /n diverges to + ∞ and (b) is larger, it diverges. 2. The radius of convergence is 1. It converges conditionally for x = 2 and absolutely for 0 < x < 2. 3. (a) Separate variables: e − x dx = e t dt . Thus- e − x = e t + C and we get C =- 2. (b) The characteristic equation has roots 1 ± i , so the general solution is x = C 1 e t cos t + C 2 e t sin t. Since x (0) = C 1 , we have C 1 = 0. Then x ′ = C 2 e t (sin t + cos t ) and, since x ′ (0) = 2, C 2 = 2. In summary, x = 2 e t sin t ....
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## This note was uploaded on 06/18/2009 for the course MATH 20D taught by Professor Mohanty during the Fall '06 term at UCSD.

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20d_04s_fs - Math 20D Final Exam Solutions June 2004 1(a...

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