Math151Quiz3Sols

Math151Quiz3Sols - QUIZ 3 (SECTIONS 11.1-11.5) SOLUTIONS...

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QUIZ 3 (SECTIONS 11.1-11.5) SOLUTIONS MATH 151 – SPRING 2004 – KUNIYUKI 105 POINTS TOTAL, BUT 100 POINTS = 100% 1) Find the limits. Write or -• when appropriate. If a limit does not exist, and and are inappropriate, write “DNE” (Does Not Exist). You do not have to show work. (8 points total; 4 points each) a) lim n n a Æ• , where a n n =+- () 41 DNE , because the terms of the sequence 3, 5, 3, 5, 3, 5, … are not approaching a single real number. b) lim n n a , where a n n =- Ê Ë Á ˆ ¯ ˜ 7 8 9 7 , because 8 9 0 Ê Ë Á ˆ ¯ ˜ Æ n as n . (Think: Geometric sequence with r == < 8 9 8 9 1.) 2) Does the series 34 2 1 n n n - Ê Ë Á ˆ ¯ ˜ = Â converge or diverge? Circle one: (You do not have to show work.) Converges Diverges (4 points) The series 11 1 12 1 n n nn = = ÂÂ = / diverges, because it is a p -series with p = 1 2 < 1. The series 1 2 1 n n = Â converges, because it is a p -series with p = 2 > 1. The given series is a linear combination of the above two series (with nonzero weights 3 and –4), so the given series diverges.
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3) Find the sum of the series 45 3 1 nn n - = () Â . (12 points) 4 5 5 4 125 5 125 4 5 125 4 5 4 5 100 4 5 3 1 3 1 1 1 1 1 1 1 n n n n n n n n n n n - = - = = = = - = - =◊ = = Ê Ë Á ˆ ¯ ˜ = Ê Ë Á ˆ ¯ ˜ Ê Ë Á ˆ ¯ ˜ = Ê Ë Á ˆ ¯ ˜ ÂÂ Â Â Â Â The sum, S , of the geometric series ar n n - = Â 1 1 r < 1 is given by S a r = - 1 . S a r = - = - = = 1 100 1 4 5 100 15 / 500 4) State the Alternating Series Test for the series - - = Â 1 1 1 n n n a , where all a n > 0 . (6 points) [The given series is clearly an alternating series.] If the a n {} sequence is nonincreasing, and if a n Æ 0 as n Æ• , then the given series converges. i.e., If 0 1 + aa kk for all k > 1, and if lim n n a = 0, then the given series converges.
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5) The series - () - = Â 1 3 1 23 1 n n n / is … (circle one:) (You do not have to show work.) Absolutely Convergent Conditionally Convergent Divergent (4 points) The series can be shown to converge by the Alternating Series Test (AST). It is an alternating series of the form - - = Â 1 1 1 n n n a , where all a n > 0. We know that 3 n / is
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Math151Quiz3Sols - QUIZ 3 (SECTIONS 11.1-11.5) SOLUTIONS...

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