official website http://uoft.me/mat137 MAT137 Calculus! Lecture 45 Today: Taylor Polynomials Taylor Series Next: Taylor Series
Power Series Definition (Power Series) A power series is a series of the form n=0 a n x n = a 0 + a 1 x + a 2 x 2 + a 3 x 3 + (1) where x is a variable and the a n s are constants called the coefficients of the series. More generally, a series of the form n=0 a n (x x 0 ) n = a 0 + a 1 (x x 0 ) + a 2 (x x 0 ) 2 + a 3 (x x 0 ) 3 + (2) is called a power series in (x x 0 ) or power series centred at x 0. Remark In equation (2), we adopt the convention (x x 0 ) 0 = 1 even when x = x 0.
Representations of Function as Power Series Example 1 Prove that is the series a n x n and b n x n both converge to the same n=1 n=1 sum for every value of x, then a n = b n for every n.
Representations of Function as Power Series Example 2 If f is a function that admits a power series representation, and it s such that f (0) = 1 and f (x) = f (x) for all x R. Find the power series representation for f.
Linear Approximation We know that if f is differentiable at x 0 the tangent line to the graph at (x 0, f (x 0 )) y = f (x 0 ) + f (x 0 )(x x 0 ) is the best linear approximation to the graph close to the point of tangency. 1 f (x) P1(x) 0.5 1 1 The are two properties that make the tangent line a good approximation close to the point of tangency: P 1 (x 0 ) = f (x 0 ) P 1 (x 0) = f (x 0 )
What is a good approximation? Definition A function g is an approximation of order n for a function f, as x x 0, when f (x) g(x) lim x x 0 (x x 0 ) n = 0 This means that the error of this approximation approaches 0 faster than (x x 0 ) n as x x 0.
Approximating Functions Theorem Assume that the functions f and g have all their derivatives. The following two statements are equivalent: 1 g is an approximation of order n for f as x x 0. In other words: f (x) g(x) lim x x 0 (x x 0 ) n = 0 2 g (x 0 ) = f (x 0 ) g (x 0 ) = f (x 0 ) g (x 0 ) = f (x 0 ) g (n) (x 0 ) = f (n) (x 0 )
Taylor Polynomials For a fucntion f with an n th order derivative at the point x 0, we may approximate f with a polynomial. Definition Let f be a function. Let x 0 R. Let n N. The n th Taylor polynomial for f at x 0 is the polynomial of smallest possible degree that is an approximation of order n for f as x x 0. Equivalently, it is the polynomial P of smallest possible degree that satisfies f (k) (a) = P (k) (a) 0 k n The Taylor series for f at x 0 is the power series S centred at x 0 that satisfies f (k) (a) = P (k) (a) k 0
Taylor Polynomials The n th Taylor polynomial for f at x 0 is a polynomial of the form P n (x) = a 0 + a 1 (x x 0 ) + a 2 (x x 0 ) 2 + a 3 (x x 0 ) 3 + + a n (x x 0 ) n such that P (k) n (x 0 ) = f (k) (x 0 ) for all 0 k n To find such a polynomial we need to find the value of the constants a 0, a 1,..., a n.
Taylor Polynomials The n th Taylor polynomial for f at x 0 is a polynomial of the form P n (x) = a 0 + a 1 (x x 0 ) + a 2 (x x 0 ) 2 + a 3 (x x 0 ) 3 + + a n (x x 0 ) n such that P (k) n (x 0 ) = f (k) (x 0 ) for all 0 k n To find such a polynomial we need to find the value of the constants a 0, a 1,..., a n. a k = f (k) (x 0 ) k! for all 0 k n
Taylor Polynomials and Taylor Series Definition (Taylor and Maclaurin Series) Let f be a function with at least n derivatives, then the n th Taylor polynomial for f at x 0 is the polynomial n k=0 f (k) (x 0 ) (x x 0 ) k k! Let f be a function with derivative of all orders on an interval (a R, a + R) for some R > 0. Then, the Taylor Series of f at x 0 (or centred at x 0 ) is n=0 f (n) (x 0 ) (x x 0 ) n n!
Taylor Polynomials and Taylor Series Example 3 Find the Taylor series for f (x) = e x at x 0 = 0.
Taylor Polynomials and Taylor Series Example 3 Find the Taylor series for f (x) = e x at x 0 = 0.
Taylor Polynomials and Taylor Series Example 4 Find the Taylor series for f (x) = sin x at x 0 = 0.
Taylor Polynomials and Taylor Series Example 4 Find the Taylor series for f (x) = sin x at x 0 = 0.
Taylor Polynomials and Taylor Series Example 4 Find the Taylor series for f (x) = sin x at x 0 = 0. Example 5 Find the Taylor series for f (x) = cos x at x 0 = 0.
Convergence of Taylor Series Recall that Definition We defined the remainder of a convergent series of constants to be the difference of the sum S of the series and a term s n from the sequence of partial sums, i.e. R n = S s n The remainder measures the error made in using s n to approximate S. Definition (The Remainder of a Function) Let f be a function with nth order derivative at every point on an open f (k) (x 0 ) interval containing x 0, and let P n (x) = n n=0 k! (x x 0 ) k, the nth Taylor polynomial for f at x 0. We define the nth remainder for f to be R n (x) = f (x) P n (x).
Convergence of Taylor Series Example 6 The 5th Taylor polynomial for f (x) = sin x at 0 is P 5 (x) = x x 3 3! + x 5 5!. Therefore, R 5 (x) = sin x (x x 3 3! + x 5 5! ).
Convergence of Taylor Series If lim n R n (x) = 0 on some interval I, then the Taylor Polynomials become better and better approximations for the function f as we include more terms of higher degree. Even more, in this case, the Taylor series is an alternative representation for f on the interval I, i.e. f (x) = n=0 f (n) (x 0 ) (x x 0 ) n for all x I n!
Convergence of Taylor Series Theorem (Taylor s Theorem) Let f be a function that can be differentiated n + 1 times in some open interval I containing x 0, and let R n (x) be the nth remainder for f at x = x 0. Then x (x t) n R n (x) = f (n+1) (t) dt n! x 0
Convergence of Taylor Series Theorem (Taylor s Theorem) Let f be a function that can be differentiated n + 1 times in some open interval I containing x 0, and let R n (x) be the nth remainder for f at x = x 0. Then x (x t) n R n (x) = f (n+1) (t) dt n! x 0 The next theorem gives us a more useful tool for understanding the remainder. Theorem (Lagrange s Form for the Remainder) Let f be a function that can be differentiated n + 1 times in some open interval I containing x 0, and let R n (x) be the nth remainder for f at x = x 0. Then for each x I, there exists c between x 0 and x such that R n (x) = f (n+1) (c) (n + 1)! (x x 0) n+1.
Convergence of Taylor Series Example 7 Show that e x = k=0 1 k! x k for all x R
Convergence of Taylor Series Example 8 Show that sin x = k=0 ( 1) k (2k + 1)! x 2k+1 for all x R