Taylor Series Math114 Department of Mathematics, University of Kentucky March 1, 2017 Math114 Lecture 18 1/ 13
Given a function, can we find a power series representation? Math114 Lecture 18 2/ 13
Given a function, can we find a power series representation? When can we find such a thing? Math114 Lecture 18 2/ 13
Given a function, can we find a power series representation? When can we find such a thing? If it exists, how can we construct it? Math114 Lecture 18 2/ 13
Suppose f can be written as a power series centered at a. Then, f (x) = c n (x a) n = c 0 + c 1 (x a) + c 2 (x a) 2 + c 3 (x a) 3 + f (x) = c 1 + 2c 2 (x a) + 3c 3 (x a) 2 + f (x) = 2c 2 + 3!c 3 (x a) + f (x) = 3!c 3 + 4!x 4 (x a) + Math114 Lecture 18 3/ 13
Suppose f can be written as a power series centered at a. Then, f (x) = c n (x a) n = c 0 + c 1 (x a) + c 2 (x a) 2 + c 3 (x a) 3 + f (x) = c 1 + 2c 2 (x a) + 3c 3 (x a) 2 + f (x) = 2c 2 + 3!c 3 (x a) + f (x) = 3!c 3 + 4!x 4 (x a) + Evaluating at x = a: f (a) = c 0 f (a) = c 1 f (a) = 2c 2 f (a) = 3!c 3. f (n) (a) = c n = c n = f (n) (a) Math114 Lecture 18 3/ 13
Theorem If f has a power series representation c n(x a) n for x a < R, then c n = f (n) (a) That is, f (x) =. f (n) (a) (x a) n for x a < R. Math114 Lecture 18 4/ 13
Theorem If f has a power series representation c n(x a) n for x a < R, then c n = f (n) (a) That is, f (x) = Defintion. f (n) (a) (x a) n for x a < R. f (n) (a) (x a) n is called the Taylor Series representation (expansion) for f centered at a. Math114 Lecture 18 4/ 13
Theorem If f has a power series representation c n(x a) n for x a < R, then c n = f (n) (a) That is, f (x) = Defintion. f (n) (a) (x a) n for x a < R. f (n) (a) (x a) n is called the Taylor Series representation (expansion) for f centered at a. If a = 0, it is sometimes called a Maclaurin Series. Math114 Lecture 18 4/ 13
Theorem If f has a power series representation c n(x a) n for x a < R, then c n = f (n) (a) That is, f (x) = Defintion. f (n) (a) (x a) n for x a < R. f (n) (a) (x a) n is called the Taylor Series representation (expansion) for f centered at a. If a = 0, it is sometimes called a Maclaurin Series. Example: Find the Maclaurin series for f (x) = e x. Math114 Lecture 18 4/ 13
Theorem If f has a power series representation c n(x a) n for x a < R, then c n = f (n) (a) That is, f (x) = Defintion. f (n) (a) (x a) n for x a < R. f (n) (a) (x a) n is called the Taylor Series representation (expansion) for f centered at a. If a = 0, it is sometimes called a Maclaurin Series. Example: Find the Maclaurin series for f (x) = e x. Note: f (n) (x) = e x for every n. Math114 Lecture 18 4/ 13
Theorem If f has a power series representation c n(x a) n for x a < R, then c n = f (n) (a) That is, f (x) = Defintion. f (n) (a) (x a) n for x a < R. f (n) (a) (x a) n is called the Taylor Series representation (expansion) for f centered at a. If a = 0, it is sometimes called a Maclaurin Series. Example: Find the Maclaurin series for f (x) = e x. Note: f (n) (x) = e x for every n. So, f (n) (0) = e 0 = 1 for every n. Math114 Lecture 18 4/ 13
Theorem If f has a power series representation c n(x a) n for x a < R, then c n = f (n) (a) That is, f (x) = Defintion. f (n) (a) (x a) n for x a < R. f (n) (a) (x a) n is called the Taylor Series representation (expansion) for f centered at a. If a = 0, it is sometimes called a Maclaurin Series. Example: Find the Maclaurin series for f (x) = e x. Note: f (n) (x) = e x for every n. So, f (n) (0) = e 0 = 1 for every n. So, the Maclaurin series for f is f (n) (0) (x 0) n 1 = x n Math114 Lecture 18 4/ 13
We have shown: IF e x has a power series representation, THEN it is xn Math114 Lecture 18 5/ 13
We have shown: IF e x has a power series representation, THEN it is xn How do we know a given function f has a power series representation? Math114 Lecture 18 5/ 13
3 2.5 2 f(x)=e x T 1 T 2 T 3 1.5 1 0.5 0-0.5-1 -0.5 0 0.5 1 Math114 Lecture 18 6/ 13
3 2.5 2 f(x)=e x T 1 T 2 T 3 1.5 1 0.5 0-0.5-1 -0.5 0 0.5 1 Math114 Lecture 18 7/ 13
3 2.5 2 f(x)=e x T 1 T 2 T 3 1.5 1 0.5 0-0.5-1 -0.5 0 0.5 1 Math114 Lecture 18 8/ 13
3 2.5 2 f(x)=e x T 1 T 2 T 3 1.5 1 0.5 0-0.5-1 -0.5 0 0.5 1 Math114 Lecture 18 9/ 13
Idea: Let T n (x) denote the n th degree Taylor polynomial. Let R n (x) = f (x) T n (x) (i.e. the remainder of the Taylor series). If lim n R n = 0, then f (x) = T n (x). Theorem If R n (x) = f (x) T n (x) and lim n R n = 0 for x a < R, then f (x) = f (n) (a) (x a) n on the interval x a < R. Math114 Lecture 18 10/ 13
Idea: Let T n (x) denote the n th degree Taylor polynomial. Let R n (x) = f (x) T n (x) (i.e. the remainder of the Taylor series). If lim n R n = 0, then f (x) = T n (x). Theorem If R n (x) = f (x) T n (x) and lim n R n = 0 for x a < R, then f (x) = f (n) (a) (x a) n on the interval x a < R. We will come back to the question: how to estimate R n? Math114 Lecture 18 10/ 13
Examples Find the Maclaurin series (i.e. the Taylor series centered at 0) for 1 f (x) = cos(x) Math114 Lecture 18 11/ 13
Examples Find the Maclaurin series (i.e. the Taylor series centered at 0) for 1 f (x) = cos(x) 2 f (x) = sin(x) Math114 Lecture 18 11/ 13
Examples Find the Maclaurin series (i.e. the Taylor series centered at 0) for 1 f (x) = cos(x) 2 f (x) = sin(x) 3 f (x) = 1 2 (ex e x ) Math114 Lecture 18 11/ 13
Examples Use series to evaluate the following limits: 1 lim x 0 sin(x) x Math114 Lecture 18 12/ 13
Examples Use series to evaluate the following limits: 1 lim x 0 sin(x) x 2 lim x 0 x ln(1 + x) x 2 Math114 Lecture 18 12/ 13
Theorem - Taylor s Remainder Estimate If f (n+1) (x) is continuous on an open interval I containing a, and x I, then there exists c [a, x] such that R n (x) = f (n+1) (c) (n+1)! (x a)n+1. Example: Using this theorem, determine how large n must be to compute e to 4 decimal places? Math114 Lecture 18 13/ 13
Theorem - Taylor s Remainder Estimate If f (n+1) (x) is continuous on an open interval I containing a, and x I, then there exists c [a, x] such that R n (x) = f (n+1) (c) (n+1)! (x a)n+1. Example: Using this theorem, determine how large n must be to compute e to 4 decimal places? In other words, how large must n be so that R n < 0.00005? Math114 Lecture 18 13/ 13