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Condensed Exam 2008

# Condensed Exam 2008 - Calculus Group I(1 Let a n> 0 for...

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Unformatted text preview: Calculus Group I (1) Let a n > 0 for all n ∈ N , and let ∞ ∑ n = 1 (- 1 ) n + 1 a n be a conditionally convergent al- ternating series. Let a + = { a 1 , a 3 , a 5 , ... } be the set of positive terms of this alternat- ing series, and let a- = {- a 2 ,- a 4 ,- a 6 , ... } be the set of negative terms of the al- ternating series. Since ∞ ∑ n = 1 (- 1 ) n + 1 a n is conditionally convergent, ∞ ∑ n = 1 a 2 n + 1 = + ∞ and ∞ ∑ n = 1- a 2 n =- ∞ . An ordered rearrangement of ∞ ∑ n = 1 (- 1 ) n + 1 a n is an infinite series which begins by adding a finite number of terms from a + in order, followed by adding a finite number of terms from a- in order, and so on. For example, one ordered rearrangement of ∞ ∑ n = 1 (- 1 ) n + 1 a n is: ( a 1 + a 3 ) + (- a 2- a 4- a 6 ) + ( a 5 + a 7 + a 9 ) + (- a 8- a 10 ) + . . . Let S > 0. Describe how to construct an ordered rearrangement of ∞ ∑ n = 1 (- 1 ) n + 1 a n whose sum is S . Calculus Group I (2) Let f ( x ) = Z x b t c dt and g ( x ) = Z 1 x b 1/ t c dt , where b t c denotes the greatest integer ≤ t . (a) Sketch the graph of f on the interval [0, 3]. (b) Find the exact value of f ( 2 2008 + 1 ) . (c) Let n denote an integer. Find lim n → ∞ g ( 1/ n ) . Is the improper integral R 1 b 1/ t c dt convergent or divergent? Justify your answer. Calculus Group I (3) Let S be the region in the plane bounded by the following curves: x = 0, y = p 6 x- x 2 , y = √ 3 3 x- 2 √ 3. Find the volume of the solid generated by rotating S about the x-axis. Calculus Group I (4) Give a MacLaurin series expansion for the function h ( x ) = x 3 tan- 1 x ....
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Condensed Exam 2008 - Calculus Group I(1 Let a n> 0 for...

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