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INFINITE SEQUENCES AND SERIES

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INFINITE SEQUENCES AND SERIES

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  1. 11 INFINITE SEQUENCES AND SERIES

  2. INFINITE SEQUENCES AND SERIES • In section 11.9, we were able to find power series representations for a certain restricted class of functions.

  3. INFINITE SEQUENCES AND SERIES • Here, we investigate more general problems. • Which functions have power series representations? • How can we find such representations?

  4. INFINITE SEQUENCES AND SERIES 11.10 Taylor and Maclaurin Series In this section, we will learn: How to find the Taylor and Maclaurin Series of a function and to multiply and divide a power series.

  5. TAYLOR & MACLAURIN SERIES Equation 1 • We start by supposing that f is any function that can be represented by a power series

  6. TAYLOR & MACLAURIN SERIES • Let’s try to determine what the coefficients cnmust be in terms of f. • To begin, notice that, if we put x = a in Equation 1, then all terms after the first one are 0 and we get: f(a) = c0

  7. TAYLOR & MACLAURIN SERIES Equation 2 • By Theorem 2 in Section 11.9, we can differentiate the series in Equation 1 term by term:

  8. TAYLOR & MACLAURIN SERIES • Substitution of x =a in Equation 2 gives: f’(a) = c1

  9. TAYLOR & MACLAURIN SERIES Equation 3 • Now, we differentiate both sides of Equation 2 and obtain:

  10. TAYLOR & MACLAURIN SERIES • Again, we put x = a in Equation 3. • The result is: f’’(a) = 2c2

  11. TAYLOR & MACLAURIN SERIES • Let’s apply the procedure one more time.

  12. TAYLOR & MACLAURIN SERIES Equation 4 • Differentiation of the series in Equation 3 gives:

  13. TAYLOR & MACLAURIN SERIES • Then, substitution of x =a in Equation 4 gives: f’’’(a) = 2 · 3c3 = 3!c3

  14. TAYLOR & MACLAURIN SERIES • By now, you can see the pattern. • If we continue to differentiate and substitute x =a, we obtain:

  15. TAYLOR & MACLAURIN SERIES • Solving the equation for the nth coefficient cn, we get:

  16. TAYLOR & MACLAURIN SERIES • The formula remains valid even for n = 0 if we adopt the conventions that 0! = 1 and f (0) = (f). • Thus, we have proved the following theorem.

  17. TAYLOR & MACLAURIN SERIES Theorem 5 • If f has a power series representation (expansion) at a, that is, if • then its coefficients are given by:

  18. TAYLOR & MACLAURIN SERIES Equation 6 • Substituting this formula for cn back into the series, we see that if f has a power series expansion at a, then it must be of the following form.

  19. TAYLOR & MACLAURIN SERIES Equation 6

  20. TAYLOR SERIES • The series in Equation 6 is called the Taylor series of the function f at a (or about a or centered at a).

  21. TAYLOR SERIES Equation 7 • For the special case a = 0, the Taylor series becomes:

  22. MACLAURIN SERIES Equation 7 • This case arises frequently enough that it is given the special name Maclaurin series.

  23. TAYLOR & MACLAURIN SERIES • The Taylor series is named after the English mathematician Brook Taylor (1685–1731). • The Maclaurin series is named for the Scottish mathematician Colin Maclaurin (1698–1746). • This is despite the fact that the Maclaurin series is really just a special case of the Taylor series.

  24. MACLAURIN SERIES • Maclaurin series are named after Colin Maclaurin because he popularized them in his calculus textbook Treatise of Fluxions published in 1742.

  25. TAYLOR & MACLAURIN SERIES Note • We have shown that if, f can be represented as a power series about a, then f is equal to the sum of its Taylor series. • However, there exist functions that are not equal to the sum of their Taylor series. • An example is given in Exercise 70.

  26. TAYLOR & MACLAURIN SERIES Example 1 • Find the Maclaurin series of the function f(x) = exand its radius of convergence.

  27. TAYLOR & MACLAURIN SERIES Example 1 • If f(x) = ex, then f (n)(x) = ex. • So,f (n)(0) = e0 = 1 for all n. • Hence, the Taylor series for f at 0 (that is, the Maclaurin series) is:

  28. TAYLOR & MACLAURIN SERIES • To find the radius of convergence, we let an =xn/n! • Then, • So, by the Ratio Test, the series converges for all x and the radius of convergence is R = ∞.

  29. TAYLOR & MACLAURIN SERIES • The conclusion we can draw from Theorem 5 and Example 1 is: • If exhas a power series expansion at 0, then

  30. TAYLOR & MACLAURIN SERIES • So, how can we determine whether exdoes have a power series representation?

  31. TAYLOR & MACLAURIN SERIES • Let’s investigate the more general question: • Under what circumstances is a function equal to the sum of its Taylor series?

  32. TAYLOR & MACLAURIN SERIES • In other words, if f has derivatives of all orders, when is the following true?

  33. TAYLOR & MACLAURIN SERIES • As with any convergent series, this means that f(x) is the limit of the sequence of partial sums.

  34. TAYLOR & MACLAURIN SERIES • In the case of the Taylor series, the partial sums are:

  35. nTH-DEGREE TAYLOR POLYNOMIAL OF f AT a • Notice that Tnis a polynomial of degree n called the nth-degree Taylor polynomial of f at a.

  36. TAYLOR & MACLAURIN SERIES • For instance, for the exponential functionf(x) = ex, the result of Example 1 shows that the Taylor polynomials at 0 (or Maclaurin polynomials) with n = 1, 2, and 3 are:

  37. TAYLOR & MACLAURIN SERIES • The graphs of the exponential function and those three Taylor polynomials are drawn here.

  38. TAYLOR & MACLAURIN SERIES • In general, f(x) is the sum of its Taylor series if:

  39. REMAINDER OF TAYLOR SERIES • If we let Rn(x) = f(x) – Tn(x) so that f(x) = Tn(x) + Rn(x) then Rn(x) is called the remainder of the Taylor series.

  40. TAYLOR & MACLAURIN SERIES • If we can somehow show that , then it follows that: • Therefore, we have proved the following.

  41. TAYLOR & MACLAURIN SERIES Theorem 8 • If f(x) = Tn(x) + Rn(x), where Tnis the nth-degree Taylor polynomial of f at aand • for |x –a| < R, then f is equal to the sum of its Taylor series on the interval |x –a| < R.

  42. TAYLOR & MACLAURIN SERIES • In trying to show that for a specific function f, we usually use the following fact.

  43. TAYLOR’S INEQUALITY Theorem 9 • If |f(n+1)(x)| ≤M for |x –a| ≤d, then the remainder Rn(x) of the Taylor series satisfies the inequality

  44. TAYLOR’S INEQUALITY • To see why this is true for n = 1, we assume that |f’’(x)| ≤M. • In particular, we have f’’(x) ≤M. • So,for a ≤ x ≤ a + d, we have:

  45. TAYLOR’S INEQUALITY • An antiderivative of f’’ is f’. • So,by Part 2 of the Fundamental Theorem of Calculus (FTC2), we have: f’(x) – f’(a) ≤M(x –a)or f’(x) ≤f’(a) + M(x –a)

  46. TAYLOR’S INEQUALITY • Thus,

  47. TAYLOR’S INEQUALITY • However, R1(x) = f(x) – T1(x) = f(x) – f(a) – f’(a)(x –a) • So,

  48. TAYLOR’S INEQUALITY • A similar argument, using f’’(x) ≥ -M, shows that: • So,

  49. TAYLOR’S INEQUALITY • We have assumed that x >a. • However,similar calculations show that this inequality is also true for x <a.

  50. TAYLOR’S INEQUALITY • This proves Taylor’s Inequality for the case where n = 1. • The result for any n is proved in a similar way by integrating n + 1 times. • See Exercise 69 for the case n = 2