Lecture of Maclaurin Taylor Series

Maclaurin and Taylor Series Instructor by Asst.Prof.Siriwan Wasukree

Maclaurin Series  We start by supposing that f is any function that can be represented by a power series: f (x)  a0  a1x  a 2 x2  a3x3  a 4 x4  . . .  a n xn  . . .

Maclaurin Series  We put x = 0 in all equation and we get:

Maclaurin Series  From a power series: f (x)  a0  a1x  a 2 x2  a3x3  a 4 x4  . . .  a n xn  . . . So, provided a power series representation for the function f(x) about x = 0,

Maclaurin Series  DEFINITIONS: Let ƒ be a function with derivatives of all orders ,then the Maclaurin series generated by ƒ at x = 0 is  f (k) (0) xk  f (0)  f (0) x  f (0) x2  f (0) x3  f (4) (0) x4  . . .  f (n) (0) xn  . . . k0 k! 1! 2! 3! 4! n!

Maclaurin Series EXAMPLE 1 Finding a Maclaurin Series of f(x)  sin x

Maclaurin Series EXAMPLE 2 Finding a Maclaurin Series of f(x)  x ex

Maclaurin Series EXAMPLE 3 Finding a Maclaurin Series of f (x)  1 1 x

Maclaurin Series

Taylor Series  The Taylor series, let’s assume that the function f(x) does in fact have a power series representation about x = a in term of (x a). Next, we will need to assume that the function f(x) has derivatives of every order and that we can in fact find them all. From the Maclaurin series:  f (k) (0) xk  f (0)  f (0) x  f (0) x2  f (0) x3  f (4) (0) x4  . . .  f (n) (0) xn  . . . k0 k! 1! 2! 3! 4! n! Changing this formula for “0” as “a” and “x” as “(x a)” into the series, we see that if f has a power series expansion at a, then it must be of the following form.

Taylor Series  This series is called the Taylor series of the function f at a (or about a or centered at a).  DEFINITIONS: Let ƒ be a function with derivatives of all orders throughout some interval containing a as an interior point. Then the Taylor series generated by ƒ at x = a is f (a)  f (a) (x  a)  f (a) (x  a)2  f (a) (x  a)3  f (4) (a) (x  a)4 1! 2! 3! 4!  f (k) (a) (x  a)k   . . .  f (n) (a) (x  a)n  . . . n! k0 k!

Taylor Series EXAMPLE 4 Find the Taylor series generated by f (x)  n x at a = 1.

Taylor Series EXAMPLE 5 Find the Taylor series for f (x)  x4 about x = 3.

Taylor Series