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23. Σ 24+1 3k-1

23. Σ 24+1 3k-1

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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help with problem 23 please, write it on paper step by step in detail, test end points(I confuse on how to do this step)

3. 4. 5. 6. 2. QUICK CHECK 5 Verify that the power series in Example 5b does not con- verge at the endpoints x = ±1.< 12. Σ SECTION 11.2 EXERCISES Review Questions Write the first four terms of a power series with coefficients Co, C1, C₂, 15, sinx 1. and c3 centered at 0. Write the first four terms of a power series with coefficients Co, C1, C2, and c3 centered at 3. What tests are used to determine the radius of convergence of a power series? = 2 ( x + Explain why a power series is tested for absolute convergence. Do the interval and radius of convergence of a power series change when the series is differentiated or integrated? Explain. What is the radius of convergence of the power series Σc, (x/2)* if the radius of convergence of Eck xk is R? What is the interval of convergence of the power series Σ(4x)^? 7. 8. How are the radii of convergence of the power series Eckx and Σ(-1)* cxk related? Basic Skills 9-28. Interval and radius of convergence Determine the radius of convergence of the following power series. Then test the endpoints to determine the interval of convergence. 9. Σ(2x) * (x - 1)* k! 2Σ k=0 2k + 1 This power series is the difference of two power series, both of which converge the interval x < 1. Therefore, by Theorem 11.4, the new series also converges If you look carefully, every example in this section is ultimately based on the p Related Exercises |x) < 1. ric series. Using this single series, we were able to develop power series for many functions. Imagine what we could do with a few more basic power series. The follow section accomplishes precisely that end. There, we discover basic power series for all standard functions of calculus. 10. (2x) k! 13. Σ(kx)* 11. Σ (x - 1)k k 3 x 2k+1 14. Σk! (x - 10)* + Σ 15. Σ sin¹ (+) 16. ²¹(x-3)* k 18. Σ(-1)- 19. Σ k²x2k 21. Σ k! 24. Σ Σ(-1) 27. Σ 29. f(3x) 5k 25. Σ (k + 1) k k20 * (2k + 1)! 28. Σ(-1)*. 29-34. Combining power series Use the geometric series 31. h(x) = 33. p(x) = = 2k f(x) Sumination notation 22. Σk (x - 1)k (x - 1)kkk = 1 1 - 3x 2x³ 1 - x 4x¹2 1- x 1 1 1 X = 17. x3k 27k 20. Σ(-1)(x x²k+1 3k-1 23. Σ 26. Σ to find the power series representation for the following function tered at 0). Give the interval of convergence of the new series. 30. g(x) for x < 1, +3 34. f(-4x) To find Give the 35. S 37. h 39. F 1- x 1 32. f(x³)=₁-x²) 1 = (-2)* (x+3) 3k+1 41-4 serie ing of c 24 1 1 + 4x 41
Transcribed Image Text:3. 4. 5. 6. 2. QUICK CHECK 5 Verify that the power series in Example 5b does not con- verge at the endpoints x = ±1.< 12. Σ SECTION 11.2 EXERCISES Review Questions Write the first four terms of a power series with coefficients Co, C1, C₂, 15, sinx 1. and c3 centered at 0. Write the first four terms of a power series with coefficients Co, C1, C2, and c3 centered at 3. What tests are used to determine the radius of convergence of a power series? = 2 ( x + Explain why a power series is tested for absolute convergence. Do the interval and radius of convergence of a power series change when the series is differentiated or integrated? Explain. What is the radius of convergence of the power series Σc, (x/2)* if the radius of convergence of Eck xk is R? What is the interval of convergence of the power series Σ(4x)^? 7. 8. How are the radii of convergence of the power series Eckx and Σ(-1)* cxk related? Basic Skills 9-28. Interval and radius of convergence Determine the radius of convergence of the following power series. Then test the endpoints to determine the interval of convergence. 9. Σ(2x) * (x - 1)* k! 2Σ k=0 2k + 1 This power series is the difference of two power series, both of which converge the interval x < 1. Therefore, by Theorem 11.4, the new series also converges If you look carefully, every example in this section is ultimately based on the p Related Exercises |x) < 1. ric series. Using this single series, we were able to develop power series for many functions. Imagine what we could do with a few more basic power series. The follow section accomplishes precisely that end. There, we discover basic power series for all standard functions of calculus. 10. (2x) k! 13. Σ(kx)* 11. Σ (x - 1)k k 3 x 2k+1 14. Σk! (x - 10)* + Σ 15. Σ sin¹ (+) 16. ²¹(x-3)* k 18. Σ(-1)- 19. Σ k²x2k 21. Σ k! 24. Σ Σ(-1) 27. Σ 29. f(3x) 5k 25. Σ (k + 1) k k20 * (2k + 1)! 28. Σ(-1)*. 29-34. Combining power series Use the geometric series 31. h(x) = 33. p(x) = = 2k f(x) Sumination notation 22. Σk (x - 1)k (x - 1)kkk = 1 1 - 3x 2x³ 1 - x 4x¹2 1- x 1 1 1 X = 17. x3k 27k 20. Σ(-1)(x x²k+1 3k-1 23. Σ 26. Σ to find the power series representation for the following function tered at 0). Give the interval of convergence of the new series. 30. g(x) for x < 1, +3 34. f(-4x) To find Give the 35. S 37. h 39. F 1- x 1 32. f(x³)=₁-x²) 1 = (-2)* (x+3) 3k+1 41-4 serie ing of c 24 1 1 + 4x 41
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