Taylor Series and Maclaurin Series

Transcription

1 Taylor Series and Maclaurin Series Definition (Taylor Series) Suppose the function f is infinitely di erentiable at a. The Taylor series of f about a (or at a or centered at a) isthepowerseries f (n) (a) (x a) n = f (a) +f 0 (a)(x a) + f 00 (a) 2! (x a) 2 + f (3) (a) (x a) ! The Taylor series about 0 is called the Maclaurin series. The definition does not address the convergence issue. The Taylor series about a is defined as long as derivatives of f of all orders exist at a. So, what s so special about the Taylor series? What is its relation with the function f? Theorem (Power series representations of functions) If f has a power series representation about a with a positive (or infinite) radius of convergence R, thenthat power series must be the Taylor series of f about a. Thus, the Taylor series is the only possible candidate for a power series representation of a function. If it turns out that the Taylor series does not represent (converge to) the function (Note: Yes, this can happen), no power series can. Math 267 (University of Calgary) Fall 205, Winter 206 / 9

2 Taylor series representation of functions Proof of the Theorem: Suppose f (x) = a n (x a) n = a 0 + a (x a) +a 2 (x a) 2 + a 3 (x a) (c R, c + R) We want to prove that the coe cients are precisely a n = f (n) (a) for all n, or, f (n) (a) =a n. Letting x = a, wehavef (a) =a 0 =0!a 0. (Recall: 0! = by definition, f (0) = f by definition) By term-by-term di erentiation, Letting x = a, wegetf 0 (a) =a =!a. Di erentiate again: f 0 (x) =a +2a 2 (x a) +3a 3 (x a) 2 +4a 4 (x a) f 00 (x) =2a a 3 (x a) +3 4a 4 (x a) a 5 (x a) Letting x = a, wegetf 00 (a) =2a 2 =2!a 2. Di erentiate again: f (3) (x) =2 3a a 4 (x a) (x a) Letting x = a, wegetf (3) (a) =2 3a 3 =3!a 3. Continue with this, we get f (n) (a) = na n = a n. Math 267 (University of Calgary) Fall 205, Winter / 9

3 Example Find the Maclaurin series of f (x) =e x.whatisitsintervalofconvergence? Solution: In a previous example, we have proved that e x x k = k! =+x + x 2 2! + x 3 3! +... ( < x < ) Since this power series represents e x with radius of convergence R =, it must be the Taylor series (about 0). The interval of convergence is (, ). Alternatively, we can compute the Maclaurin series from definition. f (n) (x) =e x for all n f (n) (0) = for all n. So, the Maclaurin series of f (x) =e x is f (n) (0) x n = x n =+x + x 2 2! + x 3 3! +... This computes the Maclaurin series but does not tell us whether the series converges to e x or not. Math 267 (University of Calgary) Fall 205, Winter / 9

4 Example Find the Maclaurin series of f (x) =sinx. Solution: We make a table: The Maclaurin series is f (n) (0) x n =0+ x +0 n f (n) (x) f (n) (0) 0 sin x 0 cos x 2 -sin x 0 3 -cos x - 4 sin x 0 5 cos x 6 -sin x 0 7 -cos x - 8 sin x 0 9 cos x... 3! x ! x ! x = ( ) k (2k +)! x 2k+. Math 267 (University of Calgary) Fall 205, Winter / 9

5 Taylor Polynomials, Taylor s Theorem To study the convergence of the Taylor series, we consider its partial sums: Definition (Taylor polynomials) The n-th order Taylor polynomial of f at a is the partial sum of the Taylor series up to the (x a) n term: nx f k (a) T n (x) = (x a) k = f (a) +f 0 (a)(x a) + f 00 (a) (x a) f (n) (a) (x k! 2! a) n Some authors call T n (x) then-th degree Taylor polynomial. However, the degree of T n (x) neednotben, sincef (n) (a) couldbeequalto0. Theorem (When is f (x) representableasapowerseries?byitstaylorseries?) Let T n (x) bethen-th order Taylor polynomial of f at a, andletr n (x) =f (x) equivalent. f can be represented by a power series about a on the open interval (a R, a + R). 2 The Taylor series of f at a converges to f (x) forallx in (a R, a + R). 3 lim R n(x) =0forallx in (a R, a + R). Proof: We have proved () =) (2). The rest are straightforward. T n (x). Then the following are Math 267 (University of Calgary) Fall 205, Winter / 9

6 Taylor s Theorem, Lagrange s form of the remainder So, the convergence issue can be resolved by analyzing the remainder term R n (x). Theorem (Taylor s Theorem) Suppose that f is n +timesdi erentiableonanopenintervali containing a. Thenforanyx in I there is a number c strictly between a and x such that R n (x) = f n+ (c) (x a)n+ (n +)! The above expression for R n (x) iscalledthelagrange s form of the remainder term. Taylor s Theorem can be proved by a method similar to the proof of the Mean Value Theorem. We will not go into the details. Corollary (One useful convergence condition) Suppose that f is infinitely di erentiable on an open interval I containing a. IfthereisanumberM such that f k (t) apple M for all t in I and for all k, then the Taylor series of f at a converges to f on I. a n+ Proof: R n (x) apple x (n +)! x k! 0 as n!. (Recall a fact: lim k! k! =0for all x.) Math 267 (University of Calgary) Fall 205, Winter / 9

7 Example Prove that e x is represented by its Maclaurin series on the interval (, ). Solution: Let f (x) =e x. Take any open interval of the form I =( A, A), where A > 0. Then for all t in I and for all k, f (k) (t) = e t = e t < e A. By our Corollary, the Maclaurin series of e x converges to e x on the interval ( A, A). Since A > 0isarbitrary,theMaclaurinseriesofe x converges to e x at all points x. e x = x k k! =+x + x 2 2! + x 3 3! +... for all x. Example Prove that the Maclaurin series of sin x converges to sin x for all x. Let f (x) =sinx and a =0. f (k) (t) isequaltooneof± sin t, ± cos t. Since ± sin t and ± cos t is between and, f (n) (t) apple. By our Corollary, the Maclaurin series of sin x converges to sin x on (, ). sin x = ( ) k (2k +)! x 2k+ = x 3! x 3 + 5! x 5 7! x for all x. Math 267 (University of Calgary) Fall 205, Winter / 9

8 Some important Maclaurin series Example Function Maclaurin series representation Radius of Conv. = ( ) k x k = x + x 2 x 3 + x x e x = x k k! =+x + x 2 2! + x 3 3! +... sin x = ( ) k (2k +)! x 2k+ = x 3! x 3 + 5! x 5 7! x cos x = ( ) k (2k)! x 2k = 2! x 2 + 4! x 4 6! x ln( + x) = k= ( ) k+ x k = x k 2 x x 3... tan x = ( ) k 2k + x 2k+ = x 3 x x 5 7 x (+x) = x k =+ x + k ( ) x 2 + 2! ( )( 2) x ! Math 267 (University of Calgary) Fall 205, Winter / 9

9 Further examples/exercises Example Verify the Maclaurin series representations on the last slide. Can you verify some of the radii of convergence? 2 Find the Taylor series of f (x) =cosx at 3. Math 267 (University of Calgary) Fall 205, Winter / 9