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Exam_exam_3_ - -1(4 5 What is the probability that...

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MIDTERM 3 — MATH 141 — FALL 2009 — BOYLE No books, no notes, no calculators. Show work on 1,2,3 and 5. Use exactly one page for each question (use the back of the page if needed). Put your name, your TA’s name and the question number on each page. Put a box around the final answer to a question. 1. (20 points) Compute the following improper integral. x =0 x e 3 x dx 2. (20 points) Compute the following indefinite integral. 1 x 2 x 2 + 1 dx 3. (20 points) Compute x x 2 + 5 x + 6 dx . 4. (15 points) For the given sequence, determine the limit. No justification required. The possible answers are a real number, + , -∞ , or DNE (does not exist). ( a ) lim n →∞ (1 + 3 /n ) n ( b ) lim n →∞ sin( n ) n ( c ) lim n →∞ 3 + 4 n + 5 n 3 6 + 7 n 3 5. (15 points) A coin is flipped until “heads” is seen. For a certain coin, the probabilty that “heads” is first seen after n flips is (1 / 5) n - 1

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Unformatted text preview: -1 (4 / 5). What is the probability that “heads” is ﬁrst seen after an odd number of ﬂips? THERE ARE MORE PROBLEMS ON THE BACK SIDE. Typeset by A M S-T E X 1 . 6.[20 points total] (i)[5 points] Let a 1 , a 2 , a 3 . . . be a sequence of real numbers, and suppose L is a real number. What is the deﬁnition for lim n →∞ a n = L ? (ii) For each of the following, write TRUE or FALSE. No justiﬁcation need be provided. [3 points each] (a) R 1 ( 1 / √ x ) dx = ∞ . (b) If lim n →∞ a n = 0, then the series ∑ ∞ n =1 a n converges. (c) A sequence of real numbers a n is bounded if a n < 100 for every n . (d) If a sequence is bounded and increasing, then it converges to a real number. (e) lim n →∞ [(1 + 1 / 2 + 1 / 3 + ··· + 1 /n )-ln( n )] = ∞ . 2...
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Exam_exam_3_ - -1(4 5 What is the probability that...

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