18.100B.MidTerm2

18.100B.MidTerm2 - 18.100B/C Midterm Exam Thursday,...

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Unformatted text preview: 18.100B/C Midterm Exam Thursday, November 13 2008, 7:309:00, 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 Problem 1. [10 points: (a) /3 (b) /7] (a) Let ( a n ) n =0 be a sequence such that a n as n . Show that the series summationdisplay n =0 ( a n a n +1 ) converges to a . (b) Show that the series summationdisplay n =0 sin(1 /n ) n converges absolutely for any > . [ Hint : Use Taylors theorem in the numerator.] Problem 2. [15 points: (a) /5 (b) /8 (c) /2] Consider the sequence ( x n ) in R defined inductively as follows: x = 2 , x n +1 = 1 2 parenleftbigg x n + 2 x n parenrightbigg ....
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18.100B.MidTerm2 - 18.100B/C Midterm Exam Thursday,...

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