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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 xaxis. Calculus Group I (4) Give a MacLaurin series expansion for the function h ( x ) = x 3 tan 1 x ....
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This note was uploaded on 09/26/2011 for the course ECE 123 taught by Professor Crank during the Spring '11 term at LSU.
 Spring '11
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