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prob10 - n n 1 Taylor Polynomials In problems 1 4 a...

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Math 151 Spring 2008 Problem Set # 10 Sequences and Taylor Polynomials Starred Problems will be turned in on Monday, April 7 Sequences In problems 1-4 list the fi rst 4 terms of the sequence { a n } . 1. a n = 2 n 1 , n = 1 , 2 , 3 , . . . 2. a n = n n 2 1 , n = 2 , 3 , 4 , . . . 3*. a n = ( 1) n 1 3 n , n = 0 , 1 , 2 , . . . 4*. a n = n 2 sin ³ (2 n + 1) π 2 ´ , n = 1 , 2 , 3 , . . . In problems 5-21, determine the fi nite or in fi nite limit, if such a limit exists. You need to display the steps that lead to your response, and provide an explanation if you claim that the limit does not exist. 5. lim n →∞ 5 n 2 + 9 2 n 2 + 1 6*. lim n →∞ n 2 + 10 4 n 3 + 1 7. lim n →∞ s 3 n 2 9 n 2 2 8*. lim n →∞ cos ( πn ) 9. lim n →∞ sin μ πn 6 n + 2 10*. lim n →∞ ( 1) n n n + 4 11. lim n →∞ ( 1) n n n 2 + 4 12*. lim n →∞ μ 3 4 n 13. lim n →∞ μ 5 3 n 14*. lim n →∞ n 3 1 2 n + 100 15*. lim n →∞ μ 3 2 n 16. lim n →∞ e n n 17*. lim n →∞ n 2 e n 18*. lim n →∞ n 2 10 n 19*. lim n →∞ ln ( n ) n 20. lim n →∞ n 1 / 3 log 10 ( n ) 21*. lim n →∞ μ 1 +
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Unformatted text preview: n ¶ n 1 Taylor Polynomials In problems 1 - 4, a) Derive the expressions for the Maclaurin polynomials of the indicated orders for f ( x ) , b) Derive the general expression for the Maclaurin polynomials for f ( x ) : 1*. f ( x ) = cos ( x ) , n = 2 , 4 , 6 2*. f ( x ) = ln ( x + 1) , n = 1 , 2 , 3 , 4 3. f ( x ) = sinh ( x ) , n = 1 , 3 , 5 4*. f ( x ) = cosh ( x ) , n = 2 , 4 , 6 In problems 5-8, derive the expressions of the Taylor polynomials of the indicated order for f ( x ) based at a : 5*. f ( x ) = arctan f ( x ) , a = 0 , n = 3 6. f ( x ) = arcsin ( x ) , a = 0 , n = 3 7. f ( x ) = √ x, a = 1 , n = 4 8*. f ( x ) = x 1 / 3 , a = 1 , n = 4 2...
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