Taylor and Maclaurin series

 Objective: Describe the procedure for finding a Taylor polynomial of a given order for a function

Introduction

Previously we discussed how to find power series representations for functions related to geometric series. Now we are interested in finding power series related to other types of functions. We address the following questions:

Which functions can be represented by power series and how do we find these representations?

If we can find a power series representation for a function f and that series converges on an interval I, how do we we prove that series actually converges to f.

Overview of Taylor and Mclaurin series 

Let's consider a function f represented by a power series at x = a. The series has the following form:

Let's determine the coefficients: c₀, c₁, c₂. Let's determine the series representation of the function f at x = a. Then the series is equal to f(a) at x = a. Let's substitute the left side by f(a) and x by a in the second side. We have: f(a) = c₀ + c₁(0-0) + c₂(0-0) + .... Then f(a) = c₀.

Let's determine the series representation of the derivative of f' at x = a. Let's first determine the derivative of f:

 

 

The series representation of f' at x = a is equal to f'(a). Let's substitute the left side by f'(a) and x by a in the right side:

f'(a) = c₁ + 2c₂.(0-0) + 3c₃(x-0)²

f'(a) = c₁

Let's determine the series representation of the second derivative of f at x = a.  Let's first determine the second derivative of f;

Let's substitute the left side by f"(a) and x by a in the right side:

f"(a) = 2c₂

c₂ = f"(a)/2

In order to determine c₃, let's determine the third derivative of f at x = a. As we know the third derivative of f has to be determined first.

Substituting the left side by f'''(a) and x by a in the right side, we have:

f'''(a) = 3.2c₃

c₃ = f'''(a)/3.2

More generally we see that if a function f has a power series representation at x = a the coefficients are given by cₙ = fⁿ(a)/n!

By substituting the coefficients in the power series representation of, we have:

This power series for f is known as the Taylor series for f a x = a. If x = 0, the series is called the series of Mclaurin of f at x = 0

Definition

Uniqueness of  the Taylor series

If a function f has a power series at a that converges to f, that series is the Taylor series of the function f at a..

Taylor polynomials

The nth partial sum of a Taylor series of a function f at a is called the nth Taylor polynomial. For example, the 0th, 1st, 2nd, 3rd Taylor polynomial of the Taylor series of the function f at a is given by:

Definition

Example

Solution

The formulas for finding the Taylor polynomials p₀, p₁, p₂, p₃ are given by:

Here a = 1. We need to find f(1), f'(1), f"(1), f"'(1) in order to find the Taylor polynomials. The function f(x) and its first, second and third derivative will allow us to find those values.

The graph of f(x) and its first three polynomials are given by:

Practice