final191 - Math 1910 Final August 5 2008 Instructions 1 Be...

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Math 1910 Final August 5, 2008 Instructions: 1. Be calm. 2. Write your name on this sheet. 3. Everything is probably easier than you think it is. 4. No robots (calculators, cell phones, etc) or notecards. 5. Algebra is your friend, you should treat it right. 6. Problems 1 – 6 are worth 14 points each, problem 7 is worth 16 points (100 in total). 7. Do all your scratch work on these pages – there are some blank ones in the back. Be sure to make it clear what your final answer is (circle, underline, or otherwise mark it). Make sure the final answer is on the same page as the problem it is answering. 8. Enjoy yourself! 1
1. (a) Does the sequence a n = 1 + 2 n e n converge? If so, find its limit. 2
(b) Does the sequence b n = ( 1) n n + 1 2 n + 1 converge? If so, find its limit. 3
2. (a) Compute cos 5 θ d θ 4
(b) Compute d θ 1 sin θ (Hint: try multiplying by 1) 5
(c) Compute dx 1 + e x 6
3. Find the radius and interval of convergence of the power series n =1 ( 1) n n ( n + 1) x n +1 7
4. In this problem, you will prove that 0 x k e x dx = k !

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