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final-99

# final-99 - Faculty of Arts and Science University of...

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Faculty of Arts and Science University of Toronto April/May Examinations MAT 137Y – Calculus! Friday, April 23, 1999 Time Alloted: 3 hours Examiners: G. Baumgartner, A. Hwang, B. Khesin, B. Madore, R. Pyke 1. For each of the series below, mark the appropriate choice, with brief justification. (A series that converges, but does not converge absolutely, is conditionally convergent .) (1a) [4 marks] n 1 n 3 1 n 2 n 1 2 Absolutely convergent Conditionally convergent Divergent (1b) [4 marks] n 1 1 n log n Absolutely convergent Conditionally convergent Divergent (1c) [4 marks] n 1 cos n 2 n 2 Absolutely convergent Conditionally convergent Divergent 2. Evaluate the following expressions; give brief justification. (2a) [5 marks] 1 0 1 x x dx (2b) [5 marks] lim x 0 1 x 2 2 cos x x 4 3. [10 marks] Find the set of x R for which the series n 1 nx n converges, and express the sum in closed form. For example,

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final-99 - Faculty of Arts and Science University of...

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