153-t2asol - Math 153 - Spring 2010 NAME Signature Test 2A...

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Math 153 - Spring 2010 NAME Signature Test 2A 21 April 2010 Answer the following questions. The answers must be clear, intelligible, and you must show your work. Provide explanation for all your steps. Your grade will be determined by adherence to these criteria. Use of books, notes and calculators is strictly forbidden. The point value of each problem is given in the left hand margin. (8 pts.) 1. Determine whether the following series converges or diverges. State clearly what test you are using and implement the test as clearly as you can. For full credit you must show how you checked all the conditions required by the hypothesis of the convergence test used. 1 X n =3 ( ± 1) n ln n n We use the Alternating series test. Consider f ( x ) = ln x x ; x ² 3 and compute f 0 ( x ) = 1 x ³ x ± ln x x 2 = 1 ± ln x x 2 Observe that 1 ± ln x < 0 if x > e so the function is eventually decreasing. Therefore b n +1 ´ b n with b n = ln n n . Next observe that lim n !1 ln n n = lim n !1 1 n = 0 by l’Hopital rule.
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153-t2asol - Math 153 - Spring 2010 NAME Signature Test 2A...

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