Compute the limit a lim x x sin 2 3 x 2 3 b lim x x 3

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13. Compute the limit (a) lim x →∞ x sin 2 3 x (b) lim x →∞ x 3 e x (c) lim x 0 e x + e - x - 2 14. Evaluate the improper integral or show it is divergent. ( a ) Z 7 0 5 ( x - 3) 4 dx ( b ) Z 0 x 2 x 2 + 5 dx ( c ) Z 6 1 dx 5 p ( x - 2) 2 ( d ) Z -∞ dx x 2 + 1 ( e ) Z 3 0 1 x dx 15. Write the series a n as a limit of partial sums.
16. Is there a number k 0 such that X n = 1 c n 2 diverges? 17. Is it true that a k = a k - 5 ? 18. Use the integral test to determine if the series n = 1 n n 2 + 1 converges or diverges. 19. Use the integral test to determine if the series n = 1 1 n (1 + (ln n ) 2 ) converges or diverges. 20. Determine if the series converges or diverges. If it converges, find the limit. (a) X n = 0 ( - 3) n 4 n + 1 Converges to 1 / 7 (b) X n = 1 2 3 n - 2 3 n + 1 ! Converges to 2 (c) X n = 1 2 - 1 3 n Diverges by n-th term divergence test (d) X n = 0 2 n + 2 5 n ! Converges to 20 / 3 (e) X n = 2 π n ( e - 1) n ! Diverges
21. Determine (using any method) whether or not the series converges or diverges.
22. Determine whether the following series converges absolutely, converges conditionally, or diverges.
23. Find the interval of convergence of the power series. (a) X n = 1 n 2 n ! x n ( -∞ , ) (b) X n = 1 ( x - 2) n n + 1 [1 , 3) (c) X n = 1 n ! x n Converges only at the center { 0 } (d) X n = 1 (2 x - 1) n n 2 [0 , 1] (e) X n = 0 - 2 x 3 n - 3 2 , 3 2
24. Derive the Taylor series for the exponential function e x centered at x = 0.
25. Find the 3th Taylor polynomial of f ( x ) = x centered at x = 1.

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