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Representation of Functions as Power Series Taylor and Maclaurin Series Definition Let f be a function such that for all n 2 N, f (n) exists on some open interval containing a. The Taylor series of f centered at a is the series 1 X f (n) (a) (x a)n . n! In the previous examples, we saw that some elementary functions can be represented using a power series (at least on some open interval). series representation PTo find a power f (x) = cn (x a)n for a function f in general, we first assume that: I I n=0 The Maclaurin series of f is the Taylor series of f centered at 0: f is infinitely di↵erentiable, that is, f has derivatives of all orders, on an open interval containing a, and; 1 X f (n) (0) n=0 f has such a power series representation. n! x n. Remark It can be shown that if f has a power series representation on some open interval containing a, then this power series must be the Taylor series. MAT1002 Week 2 13 14 Taylor Series Does Not Always Represent a Function Example (a) The Maclaurin series of f (x) := MAT1002 Week 2 ex is 1 X xn n! n=0 1 X (b) The Maclaurin series of f (x) := cos x is n=0 If a power series representation exists for an infinitely di↵erentiable function, then it must be a Taylor series. But some function may not have a power series representation in the first place. . ( 1)n x 2n . (2n)! Consider the following function and its graph. The following figure shows some partial sums of the Maclaurin series of f (x) := cos x. This function satisfies f (n) (0) = 0 for all n, so its Taylor series centered at 0 is just the zero function. Therefore f does not equal its Taylor series centered at 0 on any open interval containing 0. MAT1002 Week 2 15 MAT1002 Week 2 16 Taylor’s Theorem How do we know that a function can be represented by its Taylor series? Theorem (Taylor’s Theorem) Definition Suppose that f is a function such that f , f 0 , . . ., f (n) are continuous on [a, b], and f (n) is di↵erentiable on (a, b). Then there exists c 2 (a, b) such that ! n X f (k) (a) f (n+1) (c) k f (b) = (b a) + (b a)n+1 k! (n + 1)! f (n) Let f be a function such that for all n 2 {0, 1, . . . , N}, exists on some open interval containing a. The Taylor polynomial of f (of order n) centered at a is the polynomial Pn (x) := n X f (k) (a) k=0 k! (x a)k . k=0 = Pn (b) + Remark f (n+1) (c) (b (n + 1)! a)n+1 . A Taylor polynomial is a partial sum of a Taylor series. MAT1002 Week 2 17 f (n+1) (c) (x (n + 1)! |Rn (x)| M f (x) = Note that if Rn (x) ! 0 as n ! 1 for all x 2 I , then n!1 n!1 n=0 1 X f (n) (a) n=0 1 X f (n) (a) n! (x |x a|n+1 , (n + 1)! so Rn (x) ! 0 as n ! 1. In this case, a)n+1 for some c between a and x. f (x) = lim (Pn (x) + Rn (x)) = lim Pn (x) = 18 In particular, for a given x, if there is a positive constant M such that |f (n+1) (t)| M for all t between a and x, then Suppose that f is infinitely di↵erentiable on an open interval I containing a, and let Rn (x) := f (x) Pn (x). Then Taylor’s theorem states that Rn (x) = MAT1002 Week 2 n! (x a)n . Example a)n . (a) Show that the Maclaurin series of f (x) := e x converges to f (x) for all x 2 R. (b) Show that the Maclaurin series of f (x) := sin x converges to f (x) for all x 2 R. MAT1002 Week 2 19 MAT1002 Week 2 20 Binomial Series Definition For any m 2 R and n 2 N, define the binomial coefficient m n as follows: ✓ ◆ ✓ ◆ m m m(m 1)(m 2) . . . (m n + 1) := 1, := if n 1. 0 n n! Remark It can be shown that for x 2 ( 1, 1), (1 + x)m = 1 ✓ ◆ X m n n=0 x n. Such a series is called a binomial series. Example Find the Maclaurin series of f (x) := MAT1002 Week 2 21 Limits of Indeterminate Forms Taylor series can sometimes be used to find limits of indeterminate forms. Example Evaluate lim x!1 lnx x 1 using Taylor series. More examples can be found in Chapter 10.10 in the book. MAT1002 Week 2 23 MAT1002 Week 2 p 1 + x. 22