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Midterm Practice

# Midterm Practice - Suppose S is a non-empty subset of R...

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PRACTICE PROBLEMS FOR MIDTERM The test will be given on Monday, February 1 . It will be based on Homeworks 1-3 (Sections 1-5 and 7-8). In preparing for the test, you can practice solving the problems from the list below. In addition, take a look at the homework problems ( at least one problem on the midterm will come from the homework ), at the examples given in the textbook, and the the examples discussed in class. 1. Prove that, for any positive integer n , n k =1 k 2 k = ( n 1)2 n +1 + 2. 2. The sequence ( x n ) is deFned by the following rule: x 1 = 3, and x n +1 = 2 x n 1 for n N . Prove that, for any n N , x n = 2 n + 1. 3. Suppose x and y are positive elements of an ordered Feld, and x 2 = y 2 . Does it follow that x = y ? 4. ±ind the supremum and the inFmum of the following sets: (a) A = { 1+ n ( 1) n : n N } , (b) B = n =1 ( n 1 /n, n + 1 /n ). 5. Prove that, for every natural number n , 2 · 4 n + 1 is divisible by 3. 6.
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Unformatted text preview: Suppose S is a non-empty subset of R . Prove that there exists a sequence ( s n ) such that (i) s n ∈ S for any n ∈ N , and (ii) lim s n = sup S . Hint . Prove that, for every n ∈ N , S ∩ (sup S − 1 /n, sup S ] n = ∅ . Use this fact to select s n . 7. Determine whether the following sequences converge. Compute the limits if they exist. (a) a n = √ n 4 + 4 n − n 2 , (b) b n = (2 n + cos n ) / ( n + 1), (c) c n = ( − 1) n n 2 . 8. Prove that ( √ 5 − 1) / √ 2 is not a rational number. 9. Let a n = n/ 2 n . Prove that lim n →∞ a n = 0. Hint . You can use Exercise 1.9 (from Homework 1). 10. ±ind all rational numbers r satisfying 2 r 5 − 8 r 2 + 1 = 0. Links: the syllabus, the main page of the course. the solutions. 1...
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