Exam_exam_3_M

Exam_exam_3_M - (ii) When every term a n is a positive...

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MIDTERM 3 — MATH 141 — SPRING 2003 — BOYLE Use exactly ONE answer sheet per question (use the back of the sheet if needed). Put your name, your TA’s name and the question number on EACH page. Put a BOX around the Fnal answer to a question. No books, no notes, no calculators. Before handing in your test: on your Frst answer sheet, please copy the pledge, and sign. ———————————————————————————————————– 1. (a) (10 points) Compute the sum of the following series. X n =1 3 n 2 2 n Determine whether the following series converge or diverge. Brie±y explain. ( b ) (10 points) X n =1 sin( n 3 ) n π ( c ) (10 points) X n =1 n ( n 2 + 1) 2. (a) ²or each of the following power series, compute the radius of convergence. ( i ) (5 points) X n =1 3 n n 2 x n ( ii ) (10 points) X n =1 2 n n ! x n (c) (15 points) ²or each statement below, answer TRUE or ²ALSE. No justiFcation required. (i) The Taylor series of a polynomial is the polynomial itself.
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Unformatted text preview: (ii) When every term a n is a positive number and all partial sums of the series n =1 a n are smaller than 24, then it must be the case that the series converges. (iii) If follows from the Ratio Test that the series n =1 1 n diverges. ***THERE ARE MORE PROBLEMS ON THE BACK SIDE O THIS PAGE*** . 3. (a) (10 points ) There is a number b such that 1 + (1 / 3) + (1 / 3) 2 + (1 / 3) 3 + + (1 / 3) 200 = 1-b 1-(1 / 3) . What is that number b ? (b) (15 points) Find a positive integer n such that | e-(1 + 1 + 1 2 + 1 3! + + 1 n ! ) | < 1 10 . Explain using the Taylor Remainder Formula why your answer is correct. 4. (a) (10 points) Compute lim x 1-cos( x 4 ) x 8 . (b) (10 points) Compute f (1 / 3), if f ( x ) = x + x 2 2 + x 3 3 + x 4 4 + . . . ....
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This note was uploaded on 04/03/2010 for the course MATH 20 taught by Professor Staff during the Fall '08 term at Maryland.

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Exam_exam_3_M - (ii) When every term a n is a positive...

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