20d_04s_fe

20d_04s_fe - Math 20D Final Exam 96 points 8AM Friday June...

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Math 20D Final Exam 96 points 8AM Friday June 11, 2004 Print Name, ID number and Section on your blue book. BOOKS and CALCULATORS are NOT allowed. Both sides of one page of NOTES is allowed. You must show your work to receive credit. 1. (10 points) It can be shown that | sin( x - 1) | + | sin x | > 1 / 2 for all x . Using this fact, or otherwise, determine which of the following series converge and give reasons for your answers. (a) s n =1 2 | sin n | n 2 (b) s n =1 2 | sin n | n 2. (12 points) Consider the series s n =0 (1 - x ) n 2 n + 1 . (a) What is its radius of convergence? (b) For what values of x does it converge conditionally? (c) For what values of x does it converge absolutely? 3. (9 points per equation) Solve the following di±erential equations. If initial conditions are given, ²nd the particular solution. If no initial conditions are given, ²nd the general solution. (a)
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20d_04s_fe - Math 20D Final Exam 96 points 8AM Friday June...

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