Convergence of Taylor Series

1. Convergence of Taylor Series Objective: To find where a Taylor Series converges to the original function; approximate trig, exponential and logarithmic functions

2. The Convergence Problem • Recall that the nth Taylor polynomial for a function f about x = xo has the property that its value and the values of its first n derivatives match those of f at xo. As n increases, more and more derivatives match up, so it is reasonable to hope that for values of x near xo the values of the Taylor polynomials might converge to the value of f(x); that is

3. The Convergence Problem • However, the nth Taylor polynomial for f is the nth partial sum of the Taylor series for f, so the formula below is equivalent to stating that the Taylor series for f converges at x, and its sum is f(x).

4. The Convergence Problem • This leads to the following problem:

5. The Convergence Problem • One way to show that this is true is to show that • However, the difference appearing on the left side of this equation is the nth remainder for the Taylor series. Thus, we have the following result.

6. The Convergence Problem • Theorem:

7. Estimating the nth Remainder • It is relatively rare that one can prove directly that as . Usually, this is proved indirectly by finding appropriate bounds on and applying the Squeezing Theorem. The Remainder Estimation Theorem provides a useful bound for this purpose. Recall that this theorem asserts that if M is an upper bound for on an interval I containing xo , then

8. Example 1 • Show that the Maclaurin series for cosx converges to cosx for all x; that is

9. Example 1 • Show that the Maclaurin series for cosx converges to cosx for all x; that is • From Theorem 10.9.2, we must show that for all x as . For this purpose let f(x) = cosx, so that for all x, we have or

10. Example 1 • Show that the Maclaurin series for cosx converges to cosx for all x; that is • From Theorem 10.9.2, we must show that for all x as . For this purpose let f(x) = cosx, so that for all x, we have or • In all cases, we have so we will say that M = 1 and xo = 0 to conclude that

11. Example 1 • Show that the Maclaurin series for cosx converges to cosx for all x; that is • However, it follows that • So becomes

12. Example 1 • Show that the Maclaurin series for cosx converges to cosx for all x; that is • The following graph illustrates this point.

13. Example 2 • Use the Maclaurin series for sinx to approximate sin 3o to five decimal-point accuracy.

14. Example 2 • Use the Maclaurin series for sinx to approximate sin 3o to five decimal-point accuracy. • In the Maclaurin series • The angle is assumed to be in radians. Since 3o = p/60 it follows that

15. Example 2 • Use the Maclaurin series for sinx to approximate sin 3o to five decimal-point accuracy. • In the Maclaurin series • We must now determine how many terms in the series are required to achieve five decimal-place accuracy. We have two choices.

16. Example 2 • Use the Maclaurin series for sinx to approximate sin 3o to five decimal-point accuracy. • The Remainder Estimation Theorem. • For five decimal-place accuracy, we need • Using M = 1, x = p/60, and xo = 0, we have

17. Example 2 • Use the Maclaurin series for sinx to approximate sin 3o to five decimal-point accuracy. • The Remainder Estimation Theorem. • This happens at n = 3, so we have

18. Example 2 • Use the Maclaurin series for sinx to approximate sin 3o to five decimal-point accuracy. • The Alternating Series Test. • Let sn denote the sum of the terms up to and including the nth power of p/60. Since the exponents in the series are odd integers, the integer n must be odd, and the exponent of the first term not included in the sum sn must be n + 2.

19. Example 2 • Use the Maclaurin series for sinx to approximate sin 3o to five decimal-point accuracy. • The Alternating Series Test. • This means that for five decimal-place accuracy we must look for the first positive odd integer n such that • Again, this happens at n = 3.

20. Example 3 • Show that the Maclaurin series for ex converges to ex for all x; that is

21. Example 3 • Show that the Maclaurin series for ex converges to ex for all x; that is • Let f(x) = ex, so that • We want to show that as for all x. It will be useful to consider the cases x < 0 and x > 0 separately. If x < 0, then the interval we will look at is [x, 0] and if x > 0, the interval is [0, x].

22. Example 3 • Show that the Maclaurin series for ex converges to ex for all x; that is • Let f(x) = ex, so that • Since f n+1 (x) = ex is an increasing function, it follows that if c is in the interval [x, 0], then • If c is in the interval [0, x] then

23. Example 3 • Show that the Maclaurin series for ex converges to ex for all x; that is • We apply the Theorem with M = 1 or M = ex yielding • In both cases, the limit is 0.

24. Approximating Logarithms • The Maclaurin series is the starting point for the approximation of natural logs. Unfortunately, the usefulness of this series is limited because of its slow convergence and the restriction -1 < x < 1. However, if we replace x with –x in this series, we obtain

25. Approximating Logarithms • The Maclaurin series taking the top equation minus the bottom gives

26. Approximating Logarithms • This new series can be used to compute the natural log of any positive number y by letting or equivalently and noting that -1 < x < 1.

27. Approximating Logarithms • For example, to compute ln2 we let y = 2 in which yields x = 1/3. Substituting this value in • gives

28. Binomial Series • If m is a real number, then the Maclaurin series for (1 + x)m is called the binomial series; it is given by

29. Binomial Series • If m is a real number, then the Maclaurin series for (1 + x)m is called the binomial series; it is given by • In the case where m is a nonnegative integer, the function f(x) = (1 + x)m is a polynomial of degree m, so

30. Binomial Series • If m is a real number, then the Maclaurin series for (1 + x)m is called the binomial series; it is given by • In the case where m is a nonnegative integer, the function f(x) = (1 + x)m is a polynomial of degree m, so • The binomial series reduces to the familiar binomial expansion

31. Binomial Series • It can be proved that if m is not a nonnegative integer, then the binomial series converges to (1 + x)m if |x| < 1. Thus, for such values of x or in sigma notation

32. Example 4 • Find the binomial series for (a) (b)

33. Example 4 • Find the binomial series for (a) (b) (a) Since the general term of the binomial series is complicated, you may find it helpful to write out some of the beginning terms of the series to see developing patterns.

34. Example 4 • Find the binomial series for (a) (b) (a) Substitution m = -2 in the formula yields

35. Example 4 • Find the binomial series for (a) (b) (a) Substitution m = -2 in the formula yields

36. Example 4 • Find the binomial series for (a) (b) (b) Substitution m = -1/2 in the formula yields

37. Example 4 • Find the binomial series for (a) (b) (b) Substitution m = -1/2 in the formula yields

38. Homework • Pages 702-703 • 1, 3, 5, 9, 11, 17 • Look at page 701. There is a list of several important Maclaurin series. Be familiar with them.