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Unformatted text preview: f , using the information from this question. 5. Consider the function f x sin x . (3%) (a) Give the Taylor polynomial P 2 n 1 x of sin x of degree 2 n 1 (in powers of x ). (5%) (b) Use an estimate on the error R 2 n 1 x sin x P 2 n 1 x to ±nd a value of n such that P 2 n 1 1 5 approximates sin 1 5 with error less than 10 4 . 6. Determine, with justi±cation, whether the following series converges or diverges. (5%) (a) ∞ ∑ k 1 ln k k . 1 (5%) (b) ∞ ∑ k 1 2 k k ! k k . 7. Consider the function f x e x . (3%) (a) Write down the Taylor series (or the power series expansion in powers of x ) for f x . Include the kth (nonzero) term in your answer. (3%) (b) Express 1 e x 2 dx as a series; include the kth nonzero term in your answer. (3%) (c) Find the radius of convergence of the power series found in part (a). Justify your answer. (10%) 8. Use the εδ de±nition of limit to prove that lim x 3 x 2 9. 2...
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 Spring '10
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 Calculus, Power Series, Taylor Series, right circular cylinder, Examinations MAT 137Y

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