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133exam4solution

# 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 1 1 cos k k k = Use the limit comparison test, comparing to 1 1 k k = 2 2 1 1 cos 1 lim limcos 1 1 k k k k k k →∞ →∞ = = . Since 1 1 k k = is divergent, it follows that also 2 1 1 1 cos k k k = is divergent. (b) 2 2 1 ln k k k = Solution 1 : 2 1 ln k k 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 k k k k →∞ →∞ = = , thus 2 1 1 ln k k k = is convergent.

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133exam4solution - Math 133 Exam 4 Name Section TA...

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