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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.
 Fall '06
 Mohanty
 Math, Differential Equations, Equations

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