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Math 1910 FinalAugust 5, 2008Instructions: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. Besure to make it clear what your final answer is (circle, underline, or otherwise markit). Make sure the final answer is on the same page as the problem it is answering.8. Enjoy yourself!1
1.(a) Does the sequencean=1 + 2nenconverge? If so, find its limit.2
(b) Does the sequencebn= (−1)nn+ 12n+ 1converge? If so, find its limit.3
(b) Computedθ1−sinθ(Hint: try multiplying by 1)5
(c) Computedx1 +ex6
3. Find the radius and interval of convergence of the power series∞n=1(−1)nn(n+ 1)xn+17
4. In this problem, you will prove that∞0xke−xdx=k!