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Exam_exam_3_

# Exam_exam_3_ - n =1 n x n*THERE ARE MORE PROBLEMS ON THE...

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MIDTERM 3 — MATH 141 — FALL 2002 — 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 final answer to a question. No books, no notes, no calculators. Before handing in your test: on your first answer sheet, please copy the pledge, and sign. ———————————————————————————————————– 1. (25 points) (a) (10 points) Compute the sum of the following series. X n =1 5 1 2 · n +1 Determine whether the following series converge or diverge. Briefly explain. ( b ) X n =1 ( - 1) n p 2 + (1 /n ) ( c ) X n =1 cos 2 ( n ) n 3 / 2 ( d ) X n =1 n ( n 3 + 1) 3 / 7 2. (25 points) (a) (10 points) Find the fourth Taylor polynomial p 4 ( x ) for the function f ( x ) = 1 (5 + x 2 ) (b) For each of the following power series, compute the radius of convergence. No justification is required. ( i ) X n =1 n 5 n x n ( ii ) X n =1 1 ln( n ) · n x n ( iii )

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Unformatted text preview: n =1 n ! x n ***THERE ARE MORE PROBLEMS ON THE BACK SIDE O² THIS PAGE*** . 3. (35 points) (a) (10 points) Determine whether the following series converges, and whether it converges absolutely. Justify both answers. ∞ X n =2 (-1) n (ln( n )) 2 (b) (10) Give an appropriate upper bound for the following truncation error: f f f ∞ X n =1 1 n 2-100 X n =1 1 n 2 f f f . (c) (8) Let f ( x ) = 1 + 2 x + 3 x 2 + 4 x 3 + 5 x 4 + · · · . What is f (1 / 3)? (d) (7) Suppose you know that the power series ∑ ∞ n =1 a n x n converges at x = 3. Given this information, at which numbers in the following list can you be certain the series converges?-4 ,-3 ,-2 , , 1 4 , 2 , 4 4. (20 points) Let f ( x ) = cos( x ). (a) Find the ±fth Taylor polynomial p 5 ( x ) for f ( x ). (b) Assuming 0 < x < 2, show that | f ( x )-p 5 ( x ) | < . 1 ....
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Exam_exam_3_ - n =1 n x n*THERE ARE MORE PROBLEMS ON THE...

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