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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) sin( ) k k k 2 3 2 1 = 2 3 2 2 sin( ) 1 k k k 3 so the series converges absolutely by direct comparison with the convergent p-series 3 2 1 1 k k = 3 ( 1 2 p = > ) . Thus the series is convergent by the absolute convergence test. (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. (c) 2 1 2 ( 1) k k k k = + ! 2 2 2 1 2 ( 1) 2 ( 1)! 1 2 1 2 lim lim lim 1 0 ( 2)! 2 2 2 k k k k k k k k k k k k k k + →∞ →∞ →∞ + + + ⎞ ⎛ ⎞ ⎛ = = +

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