mid2prac - 18.100B and 18.100C Fall 2011 Second midterm...

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Fall 2011 Second midterm review sheet The second midterm covers all the material from chapters 3 – 5, and you may use results from chapters 1 – 5 without proof. Most of the midterm will consist of problems from this list. You will not be asked to reprove theorems from the book. 1. Evaluate lim n →∞ n n ! n . 2. Let p > 1. Evaluate X n =1 n - p X n =1 ( - 1) n n - p . 3. Let A > 0, and suppose X n =1 a n = A , and a n > 0 for all n N . Find the possible values of X n =1 a 2 n . 4. Let b n be a convergent series of real numbers, and let ( a n ) n N be bounded below. Prove that if a n +1 a n + b n , then ( a n ) n N converges. 5. Problem 18 from page 100. 6. Problem 25 from page 102. 7. Let X and Y be metric spaces and f a function from X to Y . Prove that the following are equivalent. (a) f is continuous. (b) If ( p n ) n N is a convergent sequence in X , then the sequence ( f ( p n )) n N is convergent in Y . (c) If (
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mid2prac - 18.100B and 18.100C Fall 2011 Second midterm...

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