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# X38 differentiation and integration of power series

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### X38 differentiation and integration of power series

1. 1. Differentiation and Integration of Power Series
2. 2. Differentiation and Integration of Power Series Theorem (Derivative and integral of Taylor series) :
3. 3. Σk=0 ∞ Differentiation and Integration of Power Series Let f(x) = P(x) = the Taylor series = ck(x – a)k for all x in the interval (a – R, a + R) where R is the radius of convergence, then Theorem (Derivative and integral of Taylor series) :
4. 4. Σk=0 ∞ Differentiation and Integration of Power Series Let f(x) = P(x) = the Taylor series = ck(x – a)k for all x in the interval (a – R, a + R) where R is the radius of convergence, then Theorem (Derivative and integral of Taylor series) : I. the Taylor series of f '(x) is P'(x) = [ck(x – a)k]' with radius of convergence R and f '(x) = P'(x) for all x in the interval (a – R, a + R). Σk=0 ∞
5. 5. Σk=0 ∞ Differentiation and Integration of Power Series Let f(x) = P(x) = the Taylor series = ck(x – a)k for all x in the interval (a – R, a + R) where R is the radius of convergence, then Theorem (Derivative and integral of Taylor series) : I. the Taylor series of f '(x) is P'(x) = [ck(x – a)k]' with radius of convergence R and f '(x) = P'(x) for all x in the interval (a – R, a + R). Σk=0 ∞ II. the Taylor series of ∫f(x)dx is ∫P(x)dx= ∫ck(x – a)kdx with radius of convergence R and ∫f(x)dx = ∫P(x)dx for all x in the interval (a – R, a + R). Σk=0 ∞
6. 6. Σk=0 ∞ Differentiation and Integration of Power Series Let f(x) = P(x) = the Taylor series = ck(x – a)k for all x in the interval (a – R, a + R) where R is the radius of convergence, then Theorem (Derivative and integral of Taylor series) : I. the Taylor series of f '(x) is P'(x) = [ck(x – a)k]' with radius of convergence R and f '(x) = P'(x) for all x in the interval (a – R, a + R). Σk=0 ∞ II. the Taylor series of ∫f(x)dx is ∫P(x)dx= ∫ck(x – a)kdx with radius of convergence R and ∫f(x)dx = ∫P(x)dx for all x in the interval (a – R, a + R). Σk=0 ∞ III. ∫ f(x)dx = ∫ ck(x – a)kdx with the series converges absolutely and (c, d) is any interval in (a – R, a + R). Σk=0 ∞ c d c d
7. 7. Example: Given the Mac series of Differentiation and Integration of Power Series Σ k=0 (2k+1)! (-1)kx2k+1∞ x – 3! x3 + 5! x5 + .. =7! x7 – Σ k=0 (2k)! (-1)kx2k∞ + 4! x4 6! x6 8! x8 +1 – – – .. =2! x2 sin(x) = cos(x) =
8. 8. Example: Given the Mac series of [sin(x)]' = Differentiation and Integration of Power Series Σ k=0 (2k+1)! (-1)kx2k+1∞ x – 3! x3 + 5! x5 + .. =7! x7 – Σ k=0 (2k)! (-1)kx2k∞ + 4! x4 6! x6 8! x8 +1 – – – .. =2! x2 sin(x) = cos(x) = The derivative of the power series of sin(x) is
9. 9. Example: Given the Mac series of [sin(x)]' = [ Differentiation and Integration of Power Series Σ k=0 (2k+1)! (-1)kx2k+1∞ x – 3! x3 + 5! x5 + .. =7! x7 – Σ k=0 (2k)! (-1)kx2k∞ + 4! x4 6! x6 8! x8 +1 – – – .. =2! x2 sin(x) = cos(x) = x – 3! x3 + 5! x5 + .. ]'7! x7 – The derivative of the power series of sin(x) is
10. 10. Example: Given the Mac series of [sin(x)]' = [ Differentiation and Integration of Power Series Σ k=0 (2k+1)! (-1)kx2k+1∞ x – 3! x3 + 5! x5 + .. =7! x7 – Σ k=0 (2k)! (-1)kx2k∞ + 4! x4 6! x6 8! x8 +1 – – – .. =2! x2 sin(x) = cos(x) = x – 3! x3 + 5! x5 + .. ]'7! x7 – + 4! x4 6! x6 + …= 1 – –2! x2 The derivative of the power series of sin(x) is
11. 11. Example: Given the Mac series of [sin(x)]' = [ Differentiation and Integration of Power Series Σ k=0 (2k+1)! (-1)kx2k+1∞ x – 3! x3 + 5! x5 + .. =7! x7 – Σ k=0 (2k)! (-1)kx2k∞ + 4! x4 6! x6 8! x8 +1 – – – .. =2! x2 sin(x) = cos(x) = x – 3! x3 + 5! x5 + .. ]'7! x7 – + 4! x4 6! x6 + …= 1 – –2! x2 = cos(x) The derivative of the power series of sin(x) is
12. 12. Example: Given the Mac series of [sin(x)]' = [ Differentiation and Integration of Power Series Σ k=0 (2k+1)! (-1)kx2k+1∞ x – 3! x3 + 5! x5 + .. =7! x7 – Σ k=0 (2k)! (-1)kx2k∞ + 4! x4 6! x6 8! x8 +1 – – – .. =2! x2 sin(x) = cos(x) = x – 3! x3 + 5! x5 + .. ]'7! x7 – + 4! x4 6! x6 + …= 1 – –2! x2 = cos(x) The derivative of the power series of sin(x) is ∫sin(x)dx = The antiderivative of the power series of sin(x) is
13. 13. Example: Given the Mac series of [sin(x)]' = [ Differentiation and Integration of Power Series Σ k=0 (2k+1)! (-1)kx2k+1∞ x – 3! x3 + 5! x5 + .. =7! x7 – Σ k=0 (2k)! (-1)kx2k∞ + 4! x4 6! x6 8! x8 +1 – – – .. =2! x2 sin(x) = cos(x) = x – 3! x3 + 5! x5 + .. ]'7! x7 – + 4! x4 6! x6 + …= 1 – –2! x2 = cos(x) The derivative of the power series of sin(x) is ∫sin(x)dx = ∫xdx – ∫ dx 3! x3 + ∫ dx5! x5 –… The antiderivative of the power series of sin(x) is
14. 14. Example: Given the Mac series of [sin(x)]' = [ Differentiation and Integration of Power Series Σ k=0 (2k+1)! (-1)kx2k+1∞ x – 3! x3 + 5! x5 + .. =7! x7 – Σ k=0 (2k)! (-1)kx2k∞ + 4! x4 6! x6 8! x8 +1 – – – .. =2! x2 sin(x) = cos(x) = x – 3! x3 + 5! x5 + .. ]'7! x7 – + 4! x4 6! x6 + …= 1 – –2! x2 = cos(x) The derivative of the power series of sin(x) is ∫sin(x)dx = ∫xdx – ∫ dx 3! x3 + ∫ dx5! x5 –… The antiderivative of the power series of sin(x) is 4! x4 6! x6 = + 2! x2 – – … + c
15. 15. Example: Given the Mac series of [sin(x)]' = [ Differentiation and Integration of Power Series Σ k=0 (2k+1)! (-1)kx2k+1∞ x – 3! x3 + 5! x5 + .. =7! x7 – Σ k=0 (2k)! (-1)kx2k∞ + 4! x4 6! x6 8! x8 +1 – – – .. =2! x2 sin(x) = cos(x) = x – 3! x3 + 5! x5 + .. ]'7! x7 – + 4! x4 6! x6 + …= 1 – –2! x2 = cos(x) The derivative of the power series of sin(x) is ∫sin(x)dx = ∫xdx – ∫ dx 3! x3 + ∫ dx5! x5 –… The antiderivative of the power series of sin(x) is 4! x4 6! x6 = + 2! x2 = -cos(x) + c– – … + c
16. 16. Example: Given the Mac series of [sin(x)]' = [ Differentiation and Integration of Power Series Σ k=0 (2k+1)! (-1)kx2k+1∞ x – 3! x3 + 5! x5 + .. =7! x7 – Σ k=0 (2k)! (-1)kx2k∞ + 4! x4 6! x6 8! x8 +1 – – – .. =2! x2 sin(x) = cos(x) = x – 3! x3 + 5! x5 + .. ]'7! x7 – + 4! x4 6! x6 + …= 1 – –2! x2 = cos(x) The derivative of the power series of sin(x) is ∫sin(x)dx = ∫xdx – ∫ dx 3! x3 + ∫ dx5! x5 –… The antiderivative of the power series of sin(x) is 4! x4 6! x6 = + 2! x2 = cos(x) + c– – … + c They all have infinite radius of convergence.
17. 17. Example: Use the fact [ ]' = Differentiation and Integration of Power Series 1 – x 1 (1 – x)2 1 to find the Mac series of . (1 – x)2 1
18. 18. Example: Use the fact [ ]' = Differentiation and Integration of Power Series 1 – x 1 (1 – x)2 1 to find the Mac series of . (1 – x)2 1 The Mac series of = 1 – x 1 Σ k=0 ∞ xk = 1 + x + x2 + x3 + ..
19. 19. Example: Use the fact [ ]' = Differentiation and Integration of Power Series 1 – x 1 (1 – x)2 1 to find the Mac series of . (1 – x)2 1 The Mac series of = 1 – x 1 Σ k=0 ∞ xk = 1 + x + x2 + x3 + .. Hence the Mac series of it's derivative is 1 – x 1 Σk=0 ∞ xk = 1 + x + x2 + x3 + ..=
20. 20. Example: Use the fact [ ]' = Differentiation and Integration of Power Series 1 – x 1 (1 – x)2 1 to find the Mac series of . (1 – x)2 1 The Mac series of = 1 – x 1 Σ k=0 ∞ xk = 1 + x + x2 + x3 + .. Hence the Mac series of it's derivative is 1 – x 1 Σk=0 ∞ xk = 1 + x + x2 + x3 + ..= (1 – x)2 1 d dx
21. 21. Example: Use the fact [ ]' = Differentiation and Integration of Power Series 1 – x 1 (1 – x)2 1 to find the Mac series of . (1 – x)2 1 The Mac series of = 1 – x 1 Σ k=0 ∞ xk = 1 + x + x2 + x3 + .. Hence the Mac series of it's derivative is Σk=0 xk[ ]' ∞ 1 – x 1 Σk=0 ∞ xk = 1 + x + x2 + x3 + ..= (1 – x)2 1 d dx d dx =
22. 22. Example: Use the fact [ ]' = Differentiation and Integration of Power Series 1 – x 1 (1 – x)2 1 to find the Mac series of . (1 – x)2 1 The Mac series of = 1 – x 1 Σ k=0 ∞ xk = 1 + x + x2 + x3 + .. Hence the Mac series of it's derivative is Σk=0 xk[ ]' = ∞ Σk=1 kxk-1 ∞ 1 – x 1 Σk=0 ∞ xk = 1 + x + x2 + x3 + ..= (1 – x)2 1 d dx d dx =
23. 23. Example: Use the fact [ ]' = Differentiation and Integration of Power Series 1 – x 1 (1 – x)2 1 to find the Mac series of . (1 – x)2 1 The Mac series of = 1 – x 1 Σ k=0 ∞ xk = 1 + x + x2 + x3 + .. Hence the Mac series of it's derivative is Σk=0 xk[ ]' = ∞ Σk=1 kxk-1 = 1 + 2x + 3x2 + 4x3 +… ∞ 1 – x 1 Σk=0 ∞ xk = 1 + x + x2 + x3 + ..= (1 – x)2 1 d dx d dx =
24. 24. Example: Use the fact [ ]' = Differentiation and Integration of Power Series 1 – x 1 (1 – x)2 1 to find the Mac series of . (1 – x)2 1 The Mac series of = 1 – x 1 Σ k=0 ∞ xk = 1 + x + x2 + x3 + .. Hence the Mac series of it's derivative is Σk=0 xk The radius of convergence of 1/(x – 1) is R = 1, [ ]' = ∞ Σk=1 kxk-1 = 1 + 2x + 3x2 + 4x3 +… ∞ 1 – x 1 Σk=0 ∞ xk = 1 + x + x2 + x3 + ..= (1 – x)2 1 d dx d dx =
25. 25. Example: Use the fact [ ]' = Differentiation and Integration of Power Series 1 – x 1 (1 – x)2 1 to find the Mac series of . (1 – x)2 1 The Mac series of = 1 – x 1 Σ k=0 ∞ xk = 1 + x + x2 + x3 + .. Hence the Mac series of it's derivative is Σk=0 xk The radius of convergence of 1/(x – 1) is R = 1, hence R =1 is the radius of convergence of [ ]' = ∞ Σk=1 kxk-1 = 1 + 2x + 3x2 + 4x3 +… ∞ Σk=1 ∞ kxk-1, 1 – x 1 Σk=0 ∞ xk = 1 + x + x2 + x3 + ..= (1 – x)2 1 d dx d dx =
26. 26. Example: Use the fact [ ]' = Differentiation and Integration of Power Series 1 – x 1 (1 – x)2 1 to find the Mac series of . (1 – x)2 1 The Mac series of = 1 – x 1 Σ k=0 ∞ xk = 1 + x + x2 + x3 + .. Hence the Mac series of it's derivative is Σk=0 xk The radius of convergence of 1/(x – 1) is R = 1, hence R =1 is the radius of convergence of and 1/(1 – x)2 = for all x in (-1, 1). [ ]' = ∞ Σk=1 kxk-1 = 1 + 2x + 3x2 + 4x3 +… ∞ Σk=1 ∞ kxk-1, Σk=1 ∞ kxk-1 1 – x 1 Σk=0 ∞ xk = 1 + x + x2 + x3 + ..= (1 – x)2 1 d dx d dx =
27. 27. We may use the integral formula to find the power series of some non-elementary functions. Differentiation and Integration of Power Series
28. 28. We may use the integral formula to find the power series of some non-elementary functions. Differentiation and Integration of Power Series Example: Find the Mac series of ∫ dxx sin(x)
29. 29. We may use the integral formula to find the power series of some non-elementary functions. Differentiation and Integration of Power Series Example: Find the Mac series of ∫ dxx sin(x) The Mac series of isx sin(x) x – 3! x3 + 5! x5 + ..7! x7 –( ) / x
30. 30. We may use the integral formula to find the power series of some non-elementary functions. Differentiation and Integration of Power Series Example: Find the Mac series of ∫ dxx sin(x) The Mac series of isx sin(x) x – 3! x3 + 5! x5 + ..7! x7 –( ) / x = 1 – 3! x2 + 5! x4 + ..7! x6 –
31. 31. We may use the integral formula to find the power series of some non-elementary functions. Differentiation and Integration of Power Series Example: Find the Mac series of ∫ dxx sin(x) The Mac series of isx sin(x) x – 3! x3 + 5! x5 + ..7! x7 –( ) / x = 1 – 3! x2 + 5! x4 + ..7! x6 – Σ k=0 (2k+1)! (-1)kx2k∞ =
32. 32. We may use the integral formula to find the power series of some non-elementary functions. Differentiation and Integration of Power Series Example: Find the Mac series of ∫ dxx sin(x) The Mac series of isx sin(x) x – 3! x3 + 5! x5 + ..7! x7 –( ) / x = 1 – 3! x2 + 5! x4 + ..7! x6 – Σ k=0 (2k+1)! (-1)kx2k∞ = So the Mac series of ∫ dx = x sin(x) Σ ∫ k=0 (2k+1)! (-1)kx2k∞ dx
33. 33. We may use the integral formula to find the power series of some non-elementary functions. Differentiation and Integration of Power Series Example: Find the Mac series of ∫ dxx sin(x) The Mac series of isx sin(x) x – 3! x3 + 5! x5 + ..7! x7 –( ) / x = 1 – 3! x2 + 5! x4 + ..7! x6 – Σ k=0 (2k+1)! (-1)kx2k∞ = So the Mac series of ∫ dx = x sin(x) Σ ∫ k=0 (2k+1)! (-1)kx2k∞ dx = Σ k=0 (2k+1)(2k+1)! (-1)kx2k+1∞
34. 34. We may use the integral formula to find the power series of some non-elementary functions. Differentiation and Integration of Power Series Example: Find the Mac series of ∫ dxx sin(x) The Mac series of isx sin(x) x – 3! x3 + 5! x5 + ..7! x7 –( ) / x = 1 – 3! x2 + 5! x4 + ..7! x6 – Σ k=0 (2k+1)! (-1)kx2k∞ = So the Mac series of ∫ dx = x sin(x) Σ ∫ k=0 (2k+1)! (-1)kx2k∞ dx = Σ k=0 (2k+1)(2k+1)! (-1)kx2k+1∞ = x – 3*3! x3 + + ..– 5*5! x5 7*7! x7
35. 35. Differentiation and Integration of Power Series Example: Find ∫ e-x dx accurate to 3 decimal places. x=0 1 2
36. 36. Differentiation and Integration of Power Series Example: Find ∫ e-x dx accurate to 3 decimal places. x=0 1 ∫ e-x dx is not an elementary function. We have to use Taylor expansion to obtain the answer we want. 2 2
37. 37. Differentiation and Integration of Power Series Example: Find ∫ e-x dx accurate to 3 decimal places. ex = n! xn x=0 1 ∫ e-x dx is not an elementary function. We have to use Taylor expansion to obtain the answer we want. 2 2 Σk=0 ∞
38. 38. Differentiation and Integration of Power Series Example: Find ∫ e-x dx accurate to 3 decimal places. ex = n! xn x=0 1 ∫ e-x dx is not an elementary function. We have to use Taylor expansion to obtain the answer we want. 2 2 Σk=0 ∞ e-x = n! (-x2)n Σk=0 ∞2 =
39. 39. Differentiation and Integration of Power Series Example: Find ∫ e-x dx accurate to 3 decimal places. ex = n! xn x=0 1 ∫ e-x dx is not an elementary function. We have to use Taylor expansion to obtain the answer we want. 2 2 Σk=0 ∞ e-x = n! (-x2)n Σk=0 ∞2 x2 + 2! x4 3! x6 4! x8 += 1 – – – ..
40. 40. Differentiation and Integration of Power Series Example: Find ∫ e-x dx accurate to 3 decimal places. ex = n! xn x=0 1 ∫ e-x dx is not an elementary function. We have to use Taylor expansion to obtain the answer we want. 2 2 Σk=0 ∞ e-x = n! (-x2)n Σk=0 ∞2 x2 + 2! x4 3! x6 4! x8 += 1 Hence ∫ e-x dx =2 n! (-x)2n Σ ∫ dx = k=0 ∞ – – – ..
41. 41. Differentiation and Integration of Power Series Example: Find ∫ e-x dx accurate to 3 decimal places. ex = n! xn x=0 1 ∫ e-x dx is not an elementary function. We have to use Taylor expansion to obtain the answer we want. 2 2 Σk=0 ∞ e-x = n! (-x2)n Σk=0 ∞2 x2 + 2! x4 3! x6 4! x8 += 1 Hence ∫ e-x dx =2 n! (-x2)n Σ ∫ dx =k=0 ∞ (2n+1)n!Σk=0 ∞ (-1)nx2n+1 – – – ..
42. 42. Differentiation and Integration of Power Series Example: Find ∫ e-x dx accurate to 3 decimal places. ex = n! xn x=0 1 ∫ e-x dx is not an elementary function. We have to use Taylor expansion to obtain the answer we want. 2 2 Σk=0 ∞ e-x = n! (-x2)n Σk=0 ∞2 x2 + 2! x4 3! x6 4! x8 += 1 Hence ∫ e-x dx =2 n! (-x2)n Σ ∫ dx =k=0 ∞ (2n+1)n!Σk=0 ∞ = c + x (-1)nx2n+1 3 x3 + 5*2! x5 + ..7*3! x7 – – – .. – –
43. 43. Differentiation and Integration of Power Series Example: Find ∫ e-x dx accurate to 3 decimal places. ex = n! xn x=0 1 ∫ e-x dx is not an elementary function. We have to use Taylor expansion to obtain the answer we want. 2 2 Σk=0 ∞ e-x = n! (-x2)n Σk=0 ∞2 x2 + 2! x4 3! x6 4! x8 += 1 Hence ∫ e-x dx =2 n! (-x2)n Σ ∫ dx =k=0 ∞ (2n+1)n!Σk=0 ∞ = c + x (-1)nx2n+1 3 x3 + 5*2! x5 + ..7*3! x7 = P(x) – – – .. – –
44. 44. Differentiation and Integration of Power Series Example: Find ∫ e-x dx accurate to 3 decimal places. ex = n! xn x=0 1 ∫ e-x dx is not an elementary function. We have to use Taylor expansion to obtain the answer we want. 2 2 Σk=0 ∞ e-x = n! (-x2)n Σk=0 ∞2 x2 + 2! x4 3! x6 4! x8 += 1 Hence ∫ e-x dx =2 n! (-x2)n Σ ∫ dx =k=0 ∞ (2n+1)n!Σk=0 ∞ = c + x (-1)nx2n+1 3 x3 + 5*2! x5 + ..7*3! x7 = P(x) ∫ e-x dx = P(1) – P(0)x=0 1 2 – – – .. – – Therefore
45. 45. Differentiation and Integration of Power Series Example: Find ∫ e-x dx accurate to 3 decimal places. ex = n! xn x=0 1 ∫ e-x dx is not an elementary function. We have to use Taylor expansion to obtain the answer we want. 2 2 Σk=0 ∞ e-x = n! (-x2)n Σk=0 ∞2 x2 + 2! x4 3! x6 4! x8 += 1 Hence ∫ e-x dx =2 n! (-x2)n Σ ∫ dx =k=0 ∞ (2n+1)n!Σk=0 ∞ = c + x (-1)nx2n+1 3 x3 + 5*2! x5 + ..7*3! x7 = P(x) ∫ e-x dx = P(1) – P(0)x=0 1 = 1 3 1 + 5*2! 1 + ..7*3! 1 2 – – – .. – – – – Therefore which is a "decreasing" alternating series.
46. 46. Differentiation and Integration of Power Series ∫ e-x dxx=0 1 = 12 Hence is a convergent alternating series. 3 1 + 5*2! 1 +7*3! 1 – – 1 9*4! – 1 11*5! + 13*6! … 1
47. 47. Differentiation and Integration of Power Series ∫ e-x dxx=0 1 = 12 Hence is a convergent alternating series. From theorem of alternaing series, any "decreasing" convergent alternating series, it's sum S satisfies |S| < |a1=1st term|. 3 1 + 5*2! 1 +7*3! 1 – – 1 9*4! – 1 11*5! + 13*6! … 1
48. 48. Differentiation and Integration of Power Series ∫ e-x dxx=0 1 = 12 Hence is a convergent alternating series. From theorem of alternaing series, any "decreasing" convergent alternating series, it's sum S satisfies |S| < |a1=1st term|. By trial and error, we find that < 0.0005.13*6! 1 3 1 + 5*2! 1 +7*3! 1 – – 1 9*4! – 1 11*5! + 13*6! … 1
49. 49. Differentiation and Integration of Power Series ∫ e-x dxx=0 1 = 12 Hence is a convergent alternating series. From theorem of alternaing series, any "decreasing" convergent alternating series, it's sum S satisfies |S| < |a1=1st term|. By trial and error, we find that < 0.0005.13*6! 1 So the tail series 1 – .. < 0.0005+ 13*6! 1 15*7! 1 17*8! – 3 1 + 5*2! 1 +7*3! 1 – – 1 9*4! – 1 11*5! + 13*6! … 1
50. 50. Differentiation and Integration of Power Series ∫ e-x dxx=0 1 = 1 3 1 + 5*2! 1 +7*3! 12 – –Hence is a convergent alternating series. From theorem of alternaing series, any "decreasing" convergent alternating series, it's sum S satisfies |S| < |a1=1st term|. By trial and error, we find that < 0.0005.13*6! 1 So the tail series 1 – .. < 0.0005+ 13*6! 1 15*7! 1 17*8! This implies the sum of the front terms 1 3 1 + 5*2! 1 +7*3! 1 – – – 1 9*4! – 1 11*5!  0.74673 is accurate to 3 decimal places. 1 9*4! – 1 11*5! + 13*6! … 1