# EXAM4 - Math 133 Exam 4 Name_ Section_ TA_ Instructions:...

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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) Using the ratio test, we find 2 2 1 k k ke = 2 2 2 2(1 ) 21 2 (1 ) 1 lim lim 1 0 1 k k k k k e k ρ −+ −− →∞ →∞ + + = < , so the series is convergent. (c) 2 1 ) 3 ! k k k k = + Using the ratio test again, we find 2 2 2 2 1 (2 ) 3 ! 3 2 3 lim lim lim 0 1 1 ) ! ) 3 1 1 1 1 k k k k k k k k k + →∞ →∞ →∞ + ++ = + + + + = < so the series is convergent.

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(d) 1 2 (1 )l n 1 k k k k = + ⎝⎠ Since 2 lim ln ln 2 0 1 k k k →∞ ⎛⎞ = ⎜⎟ + , it follows that k a
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## This note was uploaded on 05/12/2008 for the course MATH 133 taught by Professor Wei during the Spring '07 term at Michigan State University.

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EXAM4 - Math 133 Exam 4 Name_ Section_ TA_ Instructions:...

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