Subsections

• Purpose
• Getting Started
• Background
• The Ratio Test
• Radius or Interval of Convergence
• Maple Commands
• Exercises

Taylor Polynomials

Purpose

The purpose of this lab is to use Maple to introduce you to Taylor polynomial approximations to functions, including some applications.

Getting Started

To assist you, there is a worksheet associated with this lab that contains examples similar to some of the exercises. On your Maple screen go to File - Open then type the following in the white rectangle:

\\storage\academics\math\calclab\MA1023\Powerseries_start_B18.mw

You can copy the worksheet now, but you should read through the lab before you load it into Maple. Once you have read through the exercises, start up Maple, load the worksheet, and go through it carefully. Then you can start working on the exercises.

Background

The idea of the Taylor polynomial approximation of order $n$ at $x=a$, written $P_n(x,a)$, to a smooth function $f(x)$ is to require that $f(x)$ and $P_n(x,a)$ have the same value at $x=a$. Furthermore, their derivatives at $x=a$ must match up to order $n$. For example the Taylor polynomial of order three for $\sin(x)$ at $x=0$ would have to satisfy the conditions

$$\begin{array}{ccccc} P_3(0,0) & = & \sin(0) & = & 0\\ P_3'... ... & = & 0 \\ P_3'''(0,0) & = & -\cos(0) & = & -1 \end{array}$$

You should check for yourself that the cubic polynomial satisfying these four conditions is

$$P_3(x,0) = x - \frac{1}{6} x^3.$$

The general form of the Taylor polynomial approximation of order $n$ to $f(x)$ is given by the following

Theorem 1   Suppose that $f(x)$ is a smooth function in some open interval containing $x=a$. Then the $n$th degree Taylor polynomial of the function $f(x)$ at the point $x=a$ is given by

$$P_n(x,a) = \sum_{k=0}^{n} \frac{f^{(k)}(a)}{k!} (x-a)^k$$

$$= f(a) + f'(a)(x-a) + \frac{f''(a)}{2}(x-a)^2 + \cdots + \frac{f^{(n)}(a)}{n!} (x-a)^n$$

We will be seeing this formula a lot, so it would be good for you to memorize it now! The notation $f^{(k)}(a)$ is used in the definition to stand for the value of the $k$-th derivative of $f$ at $x=a$. That is, $f^{(1)}(a) = f'(a)$, $f^{(3)}(a) = f'''(a)$, and so on. By convention, $f^{(0)}(a) = f(a)$. Note that $a$ is fixed and so the derivatives $f^{(k)}(a)$ are just numbers. That is, a Taylor polynomial has the form

$$\sum_{k=0}^{n} a_k (x-a)^k$$

which you should recognize as a power series that has been truncated.

The Ratio Test

The Ratio Test for convergence of a series can be thought of as a measurement of how fast the series is increasing or decreasing. This can be found by looking at the ratio $$\frac{a_{n+1}}{an}$$ as $n \rightarrow \infty$. Given the series $$\sum_{n=0}^{\infty} a_n$$, suppose that

$$\lim_{n \rightarrow \infty} \vert\frac{a_{n+1}}{a_n}\vert=L$$

Given the series $$\sum_{n=0}^{\infty} a_n$$, suppose that

$$\lim_{n \rightarrow \infty} \vert\frac{a_{n+1}}{a_n}\vert=L$$

Then

1. the series converges if $L < 1$,
2. the series diverges if $L>1$,
3. the test is inconclusive if $L=1$.

Radius or Interval of Convergence

The radius of convergence of a series can usually be found by using the ratio test:

$$\lim_{n \rightarrow \infty} \frac{a_{n+1}}{a_n}=L$$

Next, you would need to solve for the interval of $x$ values such that $\vert L\vert<1$.

Maple Commands

To use the Taylor and TayPlot commands you need to load the CalcP7 package.

>with(CalcP7):

The exponential function can be approximated at a base point zero with a polynomial of order four using the following command.

>Taylor(exp(x),x=0,4);

You might want to experiment with changing the order. To see $f(x) = \exp(x)$ and its fourth order polynomial use

>TayPlot(exp(x),x=0,{4},x=-4..4);

This plots the exponantial and three approximating polynomials.

>TayPlot(exp(x),x=0,{2,3,4},x=-2..2);

Notice that the further away from the base point, the further the polynomial diverges from the function.

Exercises

1. Use Maple to find the interval of convergence for the following series

   a)

   
   

   $$\sum_{n=1}^{\infty} \frac{(-1)^n}{3^n}(x+2)^n$$

   
   

   b)

   
   

   $$\sum_{n=1}^{\infty} n^n x^n$$

   
   

2. Find the power series for each function listed below about $x=0$ and then determine the interval of $x$ values over which the series converges.
   1. 
      
      $$\frac{1}{1-4x}$$
      
      

   2. 
      
      $$\exp(x)$$
      
      

   3. 
      
      $$\ln(x+1)$$
      
      

3. For the function, $f(x) = 1/(3+2x)$, use the TayPlot command to plot the function and multiple Taylor polynomial approximations of various orders with base point $a=0$ on the same graph over the interval $-3 \leq x \leq 3$; use a y-range from $-5$ to $5$.

   a)

   If you increase the order of the Taylor polynomial, can you get a good approximation at $x=2$?

   b)

   Estimate the interval of convergence based on the plot and compare to the actual interval of convergence as done in the previous exercise.