133exam4solution - Math 133 Exam 4 Name_ Section_ TA_...

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Math 133 Exam 4 Name___________________ Section__________________ TA_____________________ Instructions: Please show your work. An answer alone with no supporting work will receive no credit. ________________________________________________________________________ 1.(45 pts) Classify each of the following series as either convergent or divergent. Name the test you are using, and show your work. (a) 2 1 11 cos k kk = ⎛⎞ ⎜⎟ ⎝⎠ Use the limit comparison test, comparing to 1 1 k k = 2 2 cos 1 lim lim cos 1 1 k k →∞ →∞ == . Since 1 1 k k = is divergent, it follows that also 2 1 cos k = is divergent. (b) 2 2 1 ln k = Solution 1 : 2 1 ln k 2 1 whenever so the series is convergent by comparison with the convergent p-series 3 k 2 1 1 k k = (2 . 1 p => ) Solution 2 : Compare to the convergent p-series 2 1 1 k k = using the limit comparison test: 2 2 1 1 ln lim lim 0 1 ln k k →∞ →∞ = = , thus 2 1 1 ln k = is convergent. (c) 2 1 (1 ) 3 ! k k k k = + Using the ratio test, we find
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2 2 21 2 2 1 (2 ) 3 ! 3 2 3 lim lim lim 0 1 1 (1 ) ! ) 3 1 1 1 1 k k kk k k k k k k k ρ + →∞ →∞ →∞
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This note was uploaded on 05/24/2011 for the course MTH 133 taught by Professor Staff during the Spring '08 term at Michigan State University.

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133exam4solution - Math 133 Exam 4 Name_ Section_ TA_...

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