Thomas' Calculus: Early Transcendentals

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Quiz 4 MAT104 Spring2003 (60 minutes) 1. (12 points) For each of the series below, write AC if the series converges absolutely, CC if the series converges conditionally and D if the series diverges. Please circle your answer and briefly justify it — that is, tell which tests you are using to back up your conclusions. (You do not need to work out all the details explicitly.) (a) n =1 ( - 1) n cos( ) n Ans: D Since cos( ) = ( - 1) n the numerator becomes ( - 1) 2 n = 1 so we have the harmonic series, which diverges by the p -test with p = 1. (b) n =1 ( - 1) n (ln 2) n Ans: D This series is geometric with r = - 1 / ln 2. Since 2 < e we know that ln 2 < 1. So | r | > 1 and the series diverges. (c) n =1 ( - 1) n +1 (2 n )! 5 n · n ! · n ! Ans: AC by the absolute ratio test. a n +1 a n = (2 n + 2)! 5 n +1 ( n + 1)!( n + 1)! · 5 n n ! n ! (2 n )! 4 n 2 5 n 2 = 4 5 < 1 as n → ∞ (d) n =1 ( - 1) n +1 ( n e - 1) Ans: CC by the Alternating Series Test. Since e > 1 we know that n e > n 1 = 1. So we see that this is an alternating series. Next question we would ask if whether it passes the n th term test. Here we use the Taylor expansion. Since n e = e 1 /n = 1 + 1 n + 1 2 n 2 + 1 3! n 3 + · · · then as n
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  • Fall '07
  • Nelson
  • Mathematical Series, 60 minutes

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