MATH 1231 MATHEMATICS 1B Calculus Section 4.4: Taylor & Power series.

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1 MATH 1231 MATHEMATICS 1B For use in Dr Chris Tisdell s lectures. Calculus Section 4.4: Taylor & Power series. 1. What is a Taylor series? 2. Convergence of Taylor series 3. Common Maclaurin series 4. Applications 5. What is a power series? 6. Motivation 7. Radius of convergence 8. Manipulation of power series Lecture notes created by Chris Tisdell. All images are from Thomas Calculus by Wier, Hass and Giordano, Pearson, 2008; and Calculus by Rogowski, W H Freeman & Co.,

2 1. What is a Taylor series? We now extend the idea of Taylor polynomials to Taylor series. In particular, we can now fulfil our aim of developing a method for representing a (differentiable) function f (x) as an (infinite) sum of powers of x. The main thought process behind our method is that powers of x are easy to evaluate, differentiate and integrate, so by rewriting complicated functions as sums of powers of x we can greatly simplify our analysis. Colin Maclaurin was a professor of mathematics at Edinburgh university. Newton was so impressed by Maclaurin s work that he offered to pay part of Maclaurin s salary. 2

3 Ex. Find the Maclaurin series for sin x. 3

4 Ex. Find the Maclaurin series for cos x. 4

5 Ex. Find the Maclaurin series for log(1 + x). What happens if we try to find the Maclaurin series for log x in powers of x? 5

6 Ex. Find the Taylor series for the function log x about the point x = 1. 6

7 2. Convergence. Recall from the theory of Taylor polynomials that R k (x), the remainder of order k, between a function f(x) and its Taylor polynomial P k (x) of order k is just Since R k (x) = f(x) P k (x). P (x) := n=0 f (n) (a) (x a) n n! is the limit of the partial sums P k (x) we see that the Taylor series converges to f(x) if and only if lim R k(x) = 0. (1) k The following theorem gives us a method of determining when (1) holds. 7

8 Ex: Find the Taylor series for the function cos x about the point x = π 2, (i.e. in powers of (x π 2 ).) On what interval does the Taylor series converge to cos x? 8

9 3. Common Maclaurin series. 9

10 4. Applications. This section contains some nice applications of Taylor series. We look at limits with indeterminate forms. Ex. Evaluate lim x 0 sin x x. 10

11 We now move onto definite integrals where the integrand does not have an explicit antiderivative. Ex. Express I := 1 0 sin(x 2 ) dx as an infinite series. 11

12 5. What is a power series? We have already seen two special power series: the Maclaurin series for a given function f(x) is a power series about x = 0 with in (1); a n = f (n) (0) n! the Taylor series for a given function f(x) about x = a is a power series about x = a with in (2). a n = f (n) (c) n! 12

13 6. Motivation Similar to polynomials, convergent power series can be added, subtracted, multiplied, differentiated and integrated to form new power series. Consider the function F (x) = n=0 b n x n = n=0 1 2 nxn. For what values of x does the above series converge? That is, what is the domain of F? It is clear that F (0) is defined, since substitution into the series gives F (0) = = 1. What about F (1)? Substitution into the series yields 1 F (1) = = 1 + 1/2 + 1/4 + 1/ = 2. 2n n=0 What about F (3)? We see F (3) = 1 2 n3n n=0 which is a geometric series that diverges. defined. Thus, F (3) is not Can we determine all values of x without substitution?? 13

14 7. Radius of convergence. To determine the radius of convergence, we can employ the ratio test from our earlier work on infinite series. 14

15 Let us return to our previous example of F (x) = n=0 b n x n = n=0 1 2 nxn. We compute the following ratio ρ, applying the ratio test ρ = lim b n+1 x n+1 n b n x n = lim x n+1 2 n n 2 n+1 = 1 2 lim n x = 1 2 x. By the ratio test, our series will converge when ρ = x /2 < 1. That is, when x < 2. Similarly, our series will diverge when ρ > 1, that is, when x > 2. x n 15

16 Ex. Discuss the convergence of the power series F (x) = n=0 x n n!. 16

17 Ex. Discuss the convergence of F (x) = n=1 ( 1) n n (x 5)n. 17

18 Ex. Discuss the convergence of F (x) = n=1 nx n. 18

19 8. Manipulation of power series. The following theorems state that a power series F (x) can be differentiated and integrated within its interval of convergence. We may differentiate and integrate F (x) as if it were a polynomial. 19

20 Ex: Establish a power series for log(1 + x), 1 < x < 1 by using the power series n=0 t n, 1 < t < 1. 20