]1=n n II. (a) Determine whether the series X1 n=1 ( 1)n+1 n+ 3 n3 5n+ 6 converges absolutely, converges conditionally, or diverges, and give a reason. (c) De ne the closure A of A. Office Hours: WED 8:30 – 9:30am and WED 2:30–3:30pm, or by appointment. [15pts] Let AˆR. INTRO TO REAL ANALYSIS II MIDTERM EXAM #2 April 10, 2014 This is an open book, open notes exam, with the following rules. MIT OpenCourseWare is a free & open publication of material from thousands of MIT courses, covering the entire MIT curriculum.. No enrollment or registration. If true, give a brief explanation (a complete proof is not required); if false, give a counterexample. (b) De ne what it means for x2R to be a limit point of A. (c) De ne the closure A of A. Consider the collection of subsets of X that differ from sets in Σ by a null set, (a) De ne what it means for x2R to be an isolated point of A. MATH 18.100B/C - MIDTERM 2 REVIEW 18.100B/C Real Analysis, Spring 2012 Professors: Roman Bezrukavnikov, Jared Speck I. 2.2 Proposition 1, Proposition 2, Proposition 3 . Real Analysis Math 127A-A, Winter 2019 Midterm 2: Solutions 1. This is one of over 2,200 courses on OCW. (c) Every sequence of real numbers has a monotone subsequence. Let p>1:Evaluate P 1 n=1 p n P 1 n=1 ( n1) p n III. Sometimes a sketch can help! 1. Math 323, Fall 2012 Solutions for Midterm # 2 Real Analysis I (10pts) 2. Real Analysis Math 127A-A, Winter 2019 Midterm 2: Solutions 1. You can work on the remaining problems at home. Math 6337 : Real Analysis I Mid-term Exam 2 04 November 2011 Instructions: Answer all of the problems. Please show your work! Solution: Intuitively, this series should behave like the series P 1 n=1 ( 1) n+1 1 2. At the end of class today, you must hand in your solutions to (at least) 50 points worth of problems.

Suppose that fa ng1 n=1 a sequence of strictly positive numbers and that X1 n=1 a n= A is a convergent series. (b) De ne what it means for x2R to be a limit point of A. Find materials for this course in the pages linked along the left. REAL ANALYSIS MIDTERM II 1. Solution: Intuitively, this series should behave like the series P 1 n=1 ( 1) n+1 1 2. True. MAT 341 { Applied Real Analysis Fall 2015 Midterm 2 { November 5, 2015 Solutions NAME: Please turn o your cell phone and put it away. Welcome! You are allowed to bring a note card to the exam (8.5 x 5.5in - front and back), but no other notes are allowed. to Real Analysis: Midterm Exam #2 Solutions Stephen G. Simpson Friday, March 27, 2009 1. True or False (2 points each) (a) Every monotone sequence of real numbers is convergent. Real Analysis Math 127B-B, Spring 2019 Midterm 2: Solutions 1.

State clearly and FULLY any result you are referring to. Due Friday, November 16th, by 1pm (in my mailbox) Directions: Provide complete arguments (do not skip steps).

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