adv-calc1smid - 6. Determine whether the given sequence...

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ADVANCED CALCULUS I, DR. BLOCK, SAMPLE MIDTERM, FALL 2011 PART ONE: COMPLETE THE DEFINITIONS (4 points each, 20 total points) 1. Suppose that S is a subset of the set of real numbers, and w is a real number. We say that w is an accumulation point of S if and only if 2. Suppose that f : D R , where D is a subset of R . Suppose that a is an accumulation point of D ( -∞ ,a ) . We say that lim x a - f ( x ) = if and only if 3. A sequence { a n } diverges to -∞ if and only if 4. A sequence { b n } n = i is a subsequence of the sequence { a n } n = j if and only if 5. Let { a n } be a sequence. Then lim sup n →∞ a n is PART TWO: PROOFS (1 proof, 8 total points) 5. Prove that any convergent sequence is bounded. 5 (alternate). Prove that any two limits of a convergent sequence are the same. 5 (alternate). Prove that any bounded, increasing sequence converges.
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ADVANCED CALCULUS I, DR. BLOCK, SAMPLE MIDTERM, FALL 2011 PART THREE: PROBLEMS (6 points each, 18 total points).
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Unformatted text preview: 6. Determine whether the given sequence converges or diverges. If the sequence converges, nd the limit. If the sequence diverges, determine whether the sequence diverges to ,- , or neither. Justify your answer. a n = (sin n )( n 5 n 4 + 2 n ) n 7. Evaluate the given limit if possible. Justify your answer. lim x + ( (exp( 1 x ))( x 3 + x-1) ) 8. Evaluate the given limit if possible. Justify your answer. lim x ( x sin x ) 2 3 x 4 + 2 x + 5 PART FOUR: DETERMINE IF THE STATEMENT IS TRUE OR FALSE. (2 points each, 4 total points). 10. If S is the set of rational numbers with 1 < x < 2, then 2 is an accumulation point of S . 11. Every sequence of real numbers has at least one subsequence which coverges to a real number....
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This note was uploaded on 01/03/2012 for the course MAA 4102 taught by Professor Dr.vince during the Fall '09 term at University of Florida.

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adv-calc1smid - 6. Determine whether the given sequence...

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