L29 Series

# L29 Series - 12 f 2 n ± ± 1 n 2 n g 13 f ln an = ln bn g...

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Lecture 29: Series{Some Review Prob- lems ex. State the conditions and the results of the integral test. Using the integral test to examine the problem for convergence. sin ± + 1 4 sin ± 2 + 1 9 sin ± 3 + 1 16 sin ± 4 + ±±± (con.) 1

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ex. Use comparison test to show that 2 + 3 2 3 + 4 3 3 + 5 4 3 + ±±± is convergent. 2
NYTI ex. Use ratio test to show that 1 + 1 ± 2 1 ± 3 + 1 ± 2 ± 3 1 ± 3 ± 5 + 1 ± 2 ± 3 ± 4 1 ± 3 ± 5 ± 7 + ±±± is convergent. ex. Examine each of the following series for absolute or conditional convergent or divergent: (a) 1 X n =1 ( ² 1) n +1 1 4 p n (b) 1 X n =2 ( ² 1) n 1 ln n (c) 1 ² 1 ± 3 3! + 1 ± 3 ± 5 5! + ±±± +( ² 1) n ² 1 1 ± 3 ± 5 ±±± (2 n ² 1) (2 n ² 1)! + ±±± 3

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ex. Use di±erentiation to ²nd a power series representation and radius of convergence for the functions (a) f ( x ) = x (3 ± 2 x ) 2 (b) f ( x ) = x 3 ( x ± 2) 2 4
ex. Find a power series representation and ra- dius and interval of convergence for the function f ( x ) = 1 1 + 9 x 2 Evaluate the inde±nite integral Z 1 1 + 9 x 2 dx as a power series. ex. Find the limit of the following sequences: 1. ± 1 + ( ± 1) n 1 2 n ² 2. f 1 + ( ± 1) n g 3. ± 1 + 2 n 1 + 3 n ² 4. f 1 + ( ± 2) n g 5. ± ( ± 1) n 1 ± 2 n 1 + 2 n ² 5

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6. ± ( ± 2) n + ² ± 1 2 ³ n ´ 7. n sin µ ±n 2 ¶o 8. ( sin · ±n 3 ¸ n ) 9. f p n + 1 ± p n g 10. f p n + 1( p n ± p n ± 1) g 11. n p ( n + a )( n + b ) ± p ( n ± a )( n ± b ) o

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Unformatted text preview: 12. f 2 n ± ( ± 1) n 2 n g 13. f ln( an ) = ln( bn ) g 14. n p n 2 + n ± n o 15. ¹ n p n º 16. ± n q p n ;p > 1 ´ 17. ± n n n ! ´ 6 18. ± n ! q n ² ex. For what values of p would the sequence converge? 1. f p n g where n = 0 ; 1 ; ±±± ; and p is real. 2. ± 1 n p ² 3. f (ln n ) p =n g ex. Find the sums: 1. 1 X k =1 1 (2 k ² 1)(2 k + 1) 2. 1 X k =1 2 k +1 3 k 3. 1 X k =1 2 k + 3 k 6 k 7 4. 1 2 + 1 2 3 + 1 2 5 + ±±± 5. 1 1 ± 3 + 1 3 ± 5 + 1 5 ± 7 + ±±± 6. 2 3 + 3 5 + 4 7 + 5 9 + ±±± 7. 1 2 + 1 3 + 1 4 + 1 6 + 1 8 + 1 12 + ±±± 8. 0 : 1 + 0 : 2 + 0 : 05 + 0 : 02 + 0 : 025 + 0 : 002 + ±±± 9. 1 X n =1 n + 1 n 10. 1 X n =1 5 p n + p n + 3 11. 1 X n =1 1 n ( n + 1)( n + 2) 12. 1 X n =1 n ( n + 1)! ex. . Let a 1 = p 2 ;a 2 = p 2 p 2 ;a 3 = q 2 p 2 p 2 ; ±±± ±nd the limit of a n . 8...
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L29 Series - 12 f 2 n ± ± 1 n 2 n g 13 f ln an = ln bn g...

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