18.100B.PracticeFinal

18.100B.PracticeFinal - 18.100B/C Practice Final Exam...

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Unformatted text preview: 18.100B/C Practice Final Exam Monday, December 15, 2008, 1:30–4:30, in Johnson. Closed book, no calculators. YOUR NAME: This is a 180-minute exam. No notes, books, or calculators are permitted. Point values (out of 100) are indicated for each problem. There is a (hard) bonus question, Problem 9, at the end – do not attempt it until you have worked all other problems. (Note, you can achieve the full 100 points without attempting the bonus problem.) Do all the work on these pages. GRADING 1. /10 2. /10 3. /10 4. /15 5. /10 6. /10 7. /15 8. /20 9. /20 TOTAL BONUS /100 1 Problem 1. [10 points] Suppose that x ∈ R satisfies ≤ x ≤ for every > . Show that x = 0 , using only axioms of R as an ordered field. State the axioms you are using. (Note that the Archimedean and least upper bound properties are not ordered field axioms.) 2 Problem 2. [10 points: (a) /5 (b) /5] Let ( a n ) be a sequence of positive real numbers. (a) Suppose that the series ∞ X n =1 a n converges. Prove that ∞ X n =1 √ a n a n +1 also converges.also converges....
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This note was uploaded on 12/07/2011 for the course MATH 18.100B taught by Professor Prof.katrinwehrheim during the Fall '10 term at MIT.

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18.100B.PracticeFinal - 18.100B/C Practice Final Exam...

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