practice_final - = s n =0 x n (1) Find the MacLaurin series...

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Practice Final Exam (MATH 2J: 44580) Peng Song Problem 1 (20 points) (1) Find the second degree Taylor polynomial p 2 ( x ) about the center point c = 0 for the function cos( x 2 ) using only the de±nition of the Taylor polynomial. (2) Use the polynomial from part (1) to ±nd an approximate value for the de±nite integral i 1 0 cos( x 2 ) dx Problem 2 (25 points) Find the limits of the following sequences as n → ∞ . Justify your answers. (1) a n = sin 2 ( n ) n 4 (2) b n = ne 2 n (3) c n = n n = n 1 n Problem 3 (15 points) Determine which of the following series converges. For any that converge, ±nd the limit. (1) s n =1 ln p 2 n 7 n - 5 P (2) s n =1 1 n ( n + 1) 1
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(3) s n =1 ( 2 n - 2 3 n ) Problem 4 (10 points) The MacLaurin series of the function f ( x ) = 1 1 - x is p ( x ) = 1 + x + x 2 + ··· + x n + ···
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Unformatted text preview: = s n =0 x n (1) Find the MacLaurin series of the function g ( x ) = 1 (1-x ) 2 (2) What are the radius of convergence of the MacLaurin series of the function f ( x ) and g ( x )? Problem 5 (15 points) Find the interval of the convergence J for the power series s n =1 1 n 5 n ( x-4) n and check whether this series converges or diverges at two end points of J . Problem 6 (15 points) Suppose that f ( x ) is a function which satises the equation f ( x ) =-f ( x ) for all x and f (0) = 1 (1) Determine the MacLaurin series of f ( x ). (2) What is the expression of f ( x ). 2...
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This note was uploaded on 03/11/2012 for the course MATH 2J 44360 taught by Professor Donaldson,neil during the Spring '10 term at UC Irvine.

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practice_final - = s n =0 x n (1) Find the MacLaurin series...

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