Pre-Calc Homework Solutions 394

Pre-Calc Homework Solutions 394 - 394 Chapter 9 Review an...

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Unformatted text preview: 394 Chapter 9 Review an an x 1 2n 2n 3 3 12. lim n 1 lim n 2n x 1 1 2n 1 x 12 12 1, 16. This is a geometric series with r absolutely when x2 2 1 x2 2 1 , so it converges x 3. It The series converges absolutely when x or 0 x 2. 0: 2: n 0 1, or 3 diverges for all other values of x. ( 1) ( 1) 2n 1 n 2n 1 Check x n 0 ( 1) converges 2n 1 n (a) (b) ( (c) ( 3 3, 3, 3) 3) conditionally by the Alternating Series Test. Check x n 0 ( 1)n converges conditionally by the 2n 1 (d) None 17. f (x) 1 1 x Alternating Series Test. (a) 1 (b) [0, 2] (c) (0, 2) (d) At x 13. lim n 1 x 1 . Sum 4 x2 1 x2 2 ... 1 1 4 ( 1)nx n 4 . 5 ..., evaluated at x 0 and x lim n 2 1)! x 2n 2n 1 2 18. f (x) 2n n! x 2n ln (1 x) x 2 . Sum 3 x3 3 ... 2 3 ( 1)n ln 5 . 3 1x n n , an an 1 (n evaluated at x 19. f (x) sin x x ln 1 x5 5! lim n (n 1)x 2 2 0, x , x 0 0 0. x3 3! ... sin ( 1)n 0. x 2n 1 (2n 1)! ..., evaluated at x The series converges only at x (a) 0 (b) x (c) x 0 only 0 21. f (x) 10x n 1 ln (n 1) n . Sum 1 3 x2 2! x4 4! 20. f (x) cos x ... cos 3 xn n! ( 1)n 1 . 2 x 2n (2n)! ..., evaluated at x ex 1 . Sum x2 2! (d) None 14. lim n x e ln 2 x 1 3 ... 2. ..., evaluated at an an 1 lim ln n 10x n 10x 1, x 22. f (x) ln 2. Sum tan 1 The series converges absolutely for 10x or 1 10 x x 1 . 10 1 : 10 n x3 3 x5 5 ... tan 1 ( 1)n 1 3 x 2n 1 2n 1 ..., evaluated at x 2 . Sum Check n Series Test. Check n ( 1) converges by the Alternating ln n n 6 . (Note that 1 when n is replaced by n becomes ( 1) n 1 x 2n 1 1, the general term of tan x 2n 1 , which matches the general term 1 : 10 n 1 Test, since ln n 1 10 1 1 , 10 10 1 1 , 10 10 2 1 diverges by the Direct Comparison ln n 1 1 for n 2 and diverges. n n 2 n given in the exercise.) 23. Replace x by 6x in the Maclaurin series for the end of Section 9.2. 1 1 6x 1 1 x given at (a) (b) (c) 1 1 (6x) 6x (6x)2 36x 2 ... ... (6x)n (6x)n ... ... 1 1 x 24. Replace x by x 3 in the Maclaurin series for 1 10 given at (d) At x 15. lim n the end of Section 9.2. 1 an an 1 (n 2)! x n 1 (n 1)! x n n lim lim (n n 2) x (x 0) 1 x3 1 1 (x 3) x3 (x 3) 2 x6 ... ... ( x 3)n ... ... The series converges only at x (a) 0 (b) x (c) x 0 only 0 0. ( 1)nx 3n 25. The Maclaurin series for a polynomial is the polynomial itself: 1 2x 2 x 9. (d) None ...
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