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

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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 definition of the Taylor polynomial. (2) Use the polynomial from part (1) to find an approximate value for the definite integral integraldisplay 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, find the limit. (1) summationdisplay n =1 ln parenleftbigg 2 n 7 n - 5 parenrightbigg (2) summationdisplay n =1 1 n ( n + 1) 1
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(3) summationdisplay 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
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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 satis±es 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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