MATH-138-Fall-2016-Final.pdf

A 5 x dy dx y x y 1 2016 b 5 dy dx 6 e 2 x y y 0 0

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(a) [5] x dy dx = y - x , y (1) = 2016 (b) [5] dy dx = 6 e 2 x - y , y (0) = 0 CROWDMARK MATH 138 Fall 2016 Final © 2016 University of Waterloo Please initial: Page 4 of 14
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4. Consider the sequence ( a n ) defined by a 1 = 1 and a n +1 = 1 + 2 a n for n 1. (a) [3] Prove using induction that ( a n ) is increasing. (b) [3] Prove using induction that ( a n ) is bounded above. (c) [1] State the name of the theorem that guarantees the limit lim n →∞ a n exists. (d) [3] Determine lim n →∞ a n . CROWDMARK MATH 138 Fall 2016 Final © 2016 University of Waterloo Please initial: Page 5 of 14
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5. (a) [3] State the definition of a series n =1 a n converging to a limit L . (b) [7] Prove that if a series n =1 a n is convergent, then lim n →∞ a n = 0 . CROWDMARK MATH 138 Fall 2016 Final © 2016 University of Waterloo Please initial: Page 6 of 14
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6. Determine whether each the following series is absolutely convergent, conditionally convergent or divergent. Justify your answers. (a) [5] summationdisplay n =1 cos( n ) + sin( n ) + 1 n 2 (b) [5] summationdisplay n =1 ( - 1) n sin n CROWDMARK MATH 138 Fall 2016 Final © 2016 University of Waterloo Please initial: Page 7 of 14
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(c) [5] summationdisplay n =1 5 n + 7 n 2 n + 4 n + 6 n (d) [5] summationdisplay n =1 ( - 1) n sin parenleftbigg 1 n parenrightbigg CROWDMARK MATH 138 Fall 2016 Final © 2016 University of Waterloo Please initial: Page 8 of 14
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7. Determine the radius and interval of convergence for each of the following power series. Justify
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