18.100B.MidTerm2

18.100B.MidTerm2 - 18.100B/C Midterm Exam Thursday 7:309:00...

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18.100B/C Midterm Exam Thursday, November 13 2008, 7:30–9:00, in 1-190. Closed book, no calculators. YOUR NAME: YOUR SECTION (circle one): 18.100B MWF 12-1 18.100B TR 1-2:30 18.100C This is a 90-minute evening exam. No notes, books, or calculators are permitted. Point values (out of 100) are indicated for each problem. Do all the work on these pages. GRADING 1. /10 2. /15 3. /15 4. /15 5. /10 6. /20 7. /15 TOTAL /100
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Problem 1. [10 points: (a) /3 (b) /7] (a) Let ( a n ) n =0 be a sequence such that a n 0 as n → ∞ . Show that the series summationdisplay n =0 ( a n a n +1 ) converges to a 0 . (b) Show that the series summationdisplay n =0 sin(1 /n ) n α converges absolutely for any α > 0 . [ Hint : Use Taylor’s theorem in the numerator.]
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Problem 2. [15 points: (a) /5 (b) /8 (c) /2] Consider the sequence ( x n ) in R defined inductively as follows: x 0 = 2 , x n +1 = 1 2 parenleftbigg x n + 2 x n parenrightbigg . (a) Prove that x n > 2 for all n N . (b) Show that ( x n ) is a decreasing sequence. Conclude that lim n →∞ x n exists, and compute this limit. [ Hint : the subsequence ( x n +1 ) has the same limit as ( x n ) .] (c) is ( x n
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