Math-1062 Test #2 Practice Problems (Test 2 covers 8.1, 8.2, & 11.1 - 11.8) Part I. Free Response Questions 1. Find the length of the arc of the curve for . 2. Determine whether the following sequences converge or diverge. If a sequence converges, compute its limit. A. ak (1) k 1 (2k )! (2k 1)! B. bn 4n 2 cos(n ) C. cn 9n 4 2n 3 5 4 en 3n (5 / 7) n 3. Determine whether the series converge or diverge. Justify your answers and also state the test (or combination of tests) that you use. A. n 1 E. k 2 I. n 2 cos(n ) 4n 1 B. (3) k (k 1) 2 (k !) F. 1 (2 ln n) n J. 2n 4n 3 1 n 5 6n 2 n3 C. 4m 1 ln m3 m 1 G. e n0 (1) n 2n n0 n 1 K. n k 1 6 n7 1 (n 2 1) 2 D. n3 (2) n 5n H. 2k 13k 1 kk L. n0 n0 k 1 n2 n 4n 9n n 4 8n 5n (2k )! (k!) 2 4. Apply the Integral Test to determine the convergence/divergence of the infinite series ne n2 . n 1 Verify that this series meets the conditions for using the test. 5. Find the radius and interval of convergence of the power series. State the test/tests that you use to determine the convergence/divergence of the power series at each endpoint of the interval of convergence. A. n 1 (1) n ( x 1) n n2 B. n0 (1) n 4n (2 x 1) n n 1 Part II. Multiple Choice Questions 6. Find the area of the surface obtained by rotating the curve y x3 on the interval [0, 1] about the x-axis. A. 7 (10 2 1) 27 3 B. (10 2 1) 18 1 C. D. 12 (10 1) 7 E. 14 1 7. Which of the following series is conditionally convergent? I. n 1 (1) n n (1) n n3 II. n 1 A. I only III. n 1 B. I and II only (1) n n C. I and III only D. II and III only E. I, II and III 8. Use the integral test to determine which of the following series converge? I. n 3 A. (ln n) 1 / 3 n I only 1 n (ln n) 2 II. n 3 B. II only 1 n ln n III. n 3 C. III only D. I and II only E. II and III only 9. Which of the following sequences converge? I. ak (k 1)! k! A. I only II. bn 3n cos(n ) n 3 5n B. II only III. ck C. III only (1) k k 2k 5 D. I and II only E. I, II and III 10. Which of the following series converge? I. n 1 n2 5 4n 3 3 A. II, III and IV II. n 1 4n 2 2 n 1 n 6 4n 2 1 B. I, II and IV III. 5n 2 IV. n3 n n2 C. II and IV n 1 (1) n n2 / 3 D. I and III E. I, II and III 11. Which of the following series converge? I. n 1 1 n 6 2n A. I and IV only n II. n1 1 n III. /4 B. I, II and III n 1 1 IV. n n 1 C. II, III and IV 1 nn D. I, III and IV E. I, II and IV 12. Which of the following series converge? I. m 1 m m 1 A. I and IV only II. m 1 3 4m 2 (1) (m 1) 2 B. III only m III. m0 m (1) 3m 2 1 C. II and IV only m IV. D. III and IV only m 1 (5) m 4m E. II and III only 2 13. Which of the following series converge? (1) n n en I. n 1 A. I, II and IV 14. k 1 A. II. m 1 1 m! III. n0 B. III and IV n! (1) n 5 n IV. n0 C. I, II , and III (2) n 3n D. II and III E. I, III and IV 1 3 2 k 1 3k 10 3 B. 5 3 C. 4 3 D. 25 2 E. The series diverges. 15. Consider the following two series: 3 ln n (S1) (1) n 4 n4 n n (S2) (1) n 1 n n n 1 2 Which of the following statements is true? A. B. C. D. E. S1 converges conditionally, and S2 converges conditionally. S1 converges conditionally, and S2 diverges. S1 converges absolutely, and S2 converges conditionally. S1 diverges, and S2 converges conditionally. S1 diverges, and S2 diverges. 16. Suppose n 1 an and b n 1 n are series with positive terms, and an bn for all n. Which of the following statements is not always true? A. If b n 1 n B. If b n 1 n is convergent, then (a n 1 n is convergent, then both lim an 0 and lim bn 0 . n C. If n 1 n 1 an is divergent, then E. If lim an 0 , then n (a n 1 n (1)n bn and bn is convergent, then both n 1 D. If bn ) is convergent. n (1) n 1 n an are also convergent. bn ) is also divergent. a n 1 n converges. 3 17. Assume a power series a x n0 n n converges for and diverges for . Which of the following statements is not always true? A. The radius of convergence of the power series is at least 5 and no more than 10. B. a (11) n0 n diverges. n C. a (8) n0 n n converges. D. a (4) n0 n n converges. E. We don't have enough information to determine the convergence/divergence of the series an (5) n . n0 18. Suppose that we apply the Integral Test to determine the convergence/divergence of the series n 1 1 . n2 Which of the following statements is false? A. By the Integral Test, we conclude that n 1 1 is convergent since the improper integral n2 1 1 dx is x2 convergent. B. By the Integral Test, we conclude that n 1 C. n 1 D. n2 E. 2 1 1 since the improper integral n2 1 1 1 1 2 2 ... is an over-estimate of the improper integral 2 n 2 3 1 1 1 1 2 2 2 is an under-estimate of the improper integral 2 n 2 3 4 1 1 1 1 dx 1 . x2 1 dx . x2 1 dx . x2 1 1 1 1 dx 2 2 2 ... 2 x 2 3 4 19. Which of the following statements is always true? A. If converges, then so does the series | |. B. If the terms of an alternating series decrease, then the series converges. C. If the n-th term of an alternating series is approaching 0 as n approaches infinity, then this series converges by the Alternating Series Test. D. If | | converges, then both and also converge. E. None of the above statements is true. 4 20. For which of the following series is the Ratio Test inconclusive? I. n 1 A. II only (1) n n2 II. n 1 2n n 2 n! III. B. I, II and III only n 1 n 1 n2 C. I and III only IV. n 1 n2 1 3n D. II and IV only E. III and IV only 5