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Math 2260 Exam #3 Practice Problems 1. Does the following series converge or diverge? Explain your answer. ∞ X 2n . 3n + n3 n=0 2. Does the following series converge or diverge? Explain your answer. ∞ X n . n 3 n=1 3. Does the following series converge or diverge? Explain your answer. ∞ X 1 2n sin . n n=1 4. Does the following series converge or diverge? If it converges, find the sum. If it diverges, explain why. ∞ X 2n + 3 n . 4n n=1 5. Find the interval of convergence of the power series ∞ X (2x − 5)n . n2 3 n n=1 6. Use the first two non-zero terms of an appropriate Taylor series to approximate Z 1 sin(x2 ) dx. 0 Estimate the error of your approximation (i.e. the difference between your answer and the actual value of the integral). 7. Find the radius of convergence of the Taylor series √ ∞ X (−1)n 1 + n (x − 2)n . n2 n=2 √ 8. Write the second-degree Taylor polynomial for f (x) = x centered at c = 25. Use this polynomial to √ approximate 26 and estimate the error of this approximation. (This problem is a bit computationally intensive for an actual exam problem, but it’s worth working through to see if you understand how these types of problems work.) 9. Does the series ∞ √ n X n 2 n n=1 converge or diverge. Be sure to give a complete explanation. 1 10. Does the following series converge or diverge? Explain your answer. ∞ X n!(n + 1)! . (3n)! n=1 11. Does the sequence ∞ n2 arctan n2 + 1 n=1 converge or diverge? If it converges, find the limit; if it diverges, explain why. 12. Does the series ∞ X n=2 n2 1 √ − n converge or diverge? Explain your answer. 13. Does the series ∞ X n! n5 n=1 converge or diverge? Explain your answer. 14. Does the series ∞ X 3n n3 n=1 converge or diverge? Explain your answer. 15. Does the series ∞ X 2n + 3 2 + 3n + 6)2 (n n=0 converge or diverge? Explain your answer. 16. For which values of x does the series ∞ X (x − 4)n 5n n=0 converge? What is the sum of the series when it converges? 17. Does the series ∞ X (−1)n (n2 + n3 ) n4 + 1 n=1 converge absolutely, converge conditionally, or diverge? Explain your answer. 18. Does the series ∞ X (2n)! n (n!)2 2 n=1 converge or diverge? Explain your answer. 19. Does the series ∞ X (−1)n n ln(n) n=2 converge absolutely, converge conditionally, or diverge? Explain your answer. 2 20. For what values of p does the series ∞ X n=0 (n2 1 + 1)p converge? Explain your answer. 21. What is the interval of convergence of the following power series? Explain your answer. ∞ X 3n (x − 2)n . n2 n=1 22. What are the first four nonzero terms of the Taylor series centered at x = 0 for the function f (x) = xe3x ? R 1/2 23. Which of the following gives the value of 0 cos(x2 ) dx correct to within 0.0001 (i.e. within 1/10, 000)? Explain your answer. (a) 1 2 1 1 (b) − 2 120 (c) 1 1 − 24 2688 1 1 (d) − 2 320 1 1 (e) − 2 384 24. What is the limit of the following sequence? Explain your answer. ∞ sin(arctan(ln(n))) n=2 25. Does the series √ ∞ X n=2 n2 n −n+1 converge or diverge? Explain your answer. 26. Does the series ∞ X (−1)n (n2 + n3 ) n4 + ln(n) n=1 converge absolutely, converge conditionally, or diverge? Explain your answer. 27. If it converges, find the sum of the series ∞ X (−1)n . n!2n n=0 If the series diverges, explain why. 28. For which values of p does the following series converge? Explain your answer. ∞ X √ 1 + np . n=1 29. What is the interval of convergence of the following series? Explain your answer. ∞ X (x − 1)n . n2 n=1 3 (f) 1 1 − 24 1024