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exam2prac - (a ∞ X n =0-1 n π 2 n 4 n(2 n(b ∞ X n =1...

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Math 42 — Practice Problems for Exam 2 Note: Please be sure also to check out the collection of probability questions from old exams — linked where you found this practice exam. 1. Evaluate the following integrals, showing all work. (a) Z 1 0 1 5 x dx (b) Z 2 0 1 x dx 2. Determine whether the improper integral Z 3 ln x x dx converges or diverges. 3. Determine whether the improper integral Z 1 cos 2 x x 3 dx converges or diverges. 4. [ Deleted for 2007 ] 5. True / False: (You do not need to justify your answer.) (a) Z 1 1 x 2 dx is convergent. (b) If c n 2 n is divergent, then c n ( - 3) n is divergent. (c) If a n converges, then lim n →∞ a n = 0. 6. Find the sums of the following series. (a) X n =1 1 n ( n + 2) (b) X k =1 2 k - 1 - 3 5 k +1 7. Determine whether each of the following series converges. (a) X n =1 e - 1 /n (c) X n =1 ne - n (b) X n =1 2 n ( n + 3) 3 / 2 (d) X k =1 3 k 2 + 7 8. Determine whether each of the following series converges or diverges. (a) X n =2 1 n ln n (c) X n =1 n 2 n ( n + 1) (b) X n =0 ( - 1) n n 3 + 3 n + 2 n 3 + 6
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9. Find the sums of each of the following series.
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Unformatted text preview: (a) ∞ X n =0 (-1) n π 2 n 4 n (2 n )! (b) ∞ X n =1 1-2 n 4 n 10. Find the interval of convergence of ∞ X n =1 (-2) n √ n ( x + 3) n . 11. Find a power series expansion, centered at 0, for f ( x ) = x 2 + x and its radius of convergence. 12. Isaac Newton showed that (1-x 2 )-1 / 2 = ∞ X n =0 (2 n )! 4 n ( n !) 2 x 2 n for-1 < x < 1. (a) Using this formula, find a power series expansion for arcsin x . (b) Use your power series from part (a) with x = 1 / 2 to find an infinite series whose sum is π . 13. Use power series expansions to compute lim x → e x 2-1 cos x-1 . 14. (a) Find the third-degree Taylor polynomial for f ( x ) = x 4 / 3 about a = 27. (b) Estimate the maximum error involved in estimating f with the Taylor polynomial you found in part (a) for 25 ≤ x ≤ 29....
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exam2prac - (a ∞ X n =0-1 n π 2 n 4 n(2 n(b ∞ X n =1...

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