Math191_Week15_section

# Math191_Week15_section - x does the series converge(b...

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Math 191 Problem Set for Week 13 Section Theorem 0.1 (The nth Term Test) If a n fails to exist or is diﬀerent from zero, then X n =1 a n diverges. Theorem 0.2 (The Alternating Series Test) The series X n =1 ( - 1) n +1 u n = u 1 - u 2 + u 3 - u 4 + ... (1) converges if all three of the following conditions are satisﬁed: a. The u n ’s are all positive. b. u n u n +1 for all n N , for some integer N . c. u n 0 as n → ∞ . Deﬁnition 0.3 (Absolutely Convergent) A series a n converges absolutely if the corresponding series of absolutel values | a n | converges. Deﬁnition 0.4 (Conditionally Convergent) A series that converges but does not converge absolutely con- verges conditionally. Theorem 0.5 (The Absolute Convergence Test) If X n =1 | a n | converges, then X n =1 a n converges. Which of the series in the following converge absolutely, or conditionally, and which diverge? 1. X n =1 ( - 1) n +1 1 n 2 2. X n =2 ( - 1) n +1 1 ln n 3. X n =1 ( - 1) n +1 3 + n 5 + n 4. X n =1 ( - 1) n +1 tan - 1 n n 2 + 1 5. X n =1 ( - 1) n +1 ( q n + n - n ) 6. In the series X n =1 (1 + 1 n ) n x n , (a) ﬁnd the series’ radius and interval of convergence. For what values of

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Unformatted text preview: x does the series converge (b) absolutely, (c) conditionally? Deﬁnition 0.6 (Taylor and Maclaurin series) Let f be a function with derivatives of all orders through-out some interval containing a as an interior point. Then Taylor series generated by f at x = a is ∞ X k =0 f ( k ) ( a ) k ! ( x-a ) k (2) The Maclaurin series generated by f is ∞ X k =0 f ( k ) (0) k ! x k , the Taylor series generated by f at x = 0 . 1 Deﬁnition 0.7 (Taylor Polynomial of order n ) displaystyleP n ( x ) = ∑ n k =0 f ( k ) ( a ) k ! ( x-a ) k 6. Find the Taylor polynomials of orders , 1 , 2 and 3 . a. f ( x ) = 1 x , a = 2 b. f ( x ) = sin x, a = π 4 7. Find the Malaurin series for a. e-x b. sin 3 x 8. Find the Taylor series a. f ( x ) = 1 x 2 , a = 1 b. f ( x ) = x 1-x , a = 0 2...
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Math191_Week15_section - x does the series converge(b...

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