prelim1_311_sol

prelim1_311_sol - Mathematics 311 Preliminary Exam 1...

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Unformatted text preview: Mathematics 311 Preliminary Exam 1 Introduction to Analysis Name : Instructor: Marius Ionescu Instructions: 1. Please write your name in the blank above, and sign & date below. 2. Please use the space provided to write your solution. 3. If you need extra pages please attach them at the end and clearly indicate each problem number where your solution begins. 4. Please staple your additional pages together in order with these pages on top. 5. This prelim is due by Wednesday, March 5, at 5:00pm. You may submit it during Wednesday morning’s class or Wednesday afternoon in my office. TIP: The exam will be significantly easier (and shorter) if you cite results (e.g., theorems and examples covered in lecture), rather than proving them from scratch. If you are unsure whether or not you are allowed to use certain things, please contact me. This exam contains 8 questions, for a total of 100 points. Good luck! You may use your notes, results proved in the textbook, and results that you have proved in homework assignments (provided they are correct). You may use work done with others on past assignments, and work done in office hours with the instructor or the TA. Beyond this however, exam solutions must be entirely your own work. Upon receipt of this exam, you are on your honour not to discuss any material covered on this exam with anyone other than the instructor. Please sign below to indicate that you understand and agree: Signature: Date: Mathematics 311 Preliminary Exam 1 Introduction to Analysis 1 /20 2 /5 3 /5 4 /15 5 /10 6 /20 7 /15 8 /10 T /100 Page 2 of 10 Please go on to the next page ... Mathematics 311 Preliminary Exam 1 Introduction to Analysis 1. Give an example of each of the following, or use theorems to prove that such a thing is impossible. (a) (4pts) Sequences { a n } , { b n } which both diverge, but whose sum { a n + b n } converges. Solution: If a n = (- 1) n and b n = (- 1) n +1 , then both sequences are divergent (we proved this in class). Their sum, however, equals 0 for all n . Thus a n + b n is convergent to 0....
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This note was uploaded on 09/29/2011 for the course MATH 3110 at Cornell.

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prelim1_311_sol - Mathematics 311 Preliminary Exam 1...

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