Subsections

• Introduction
• Getting Started
• Background
• The Ratio Test
• Radius or Interval of Convergence
• Exercises

Sequences and Series

Introduction

The purpose of this lab is to learn how to define sequences and series using Maple as well as observe their plots and test for convergence.

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\Series_start_C16.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 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

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

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 $L < 1$. Below is an example of how you could use Maple to find the interval of convergence for the power series for the function $$f(x)=\frac{x}{3-x^2}$$.

> series(x/(3-x^2),x=0,12);
> a:=n->x^(2*n+1)/3^(n+1);
> l:=simplify(limit(abs(a(n+1)/a(n)),n=infinity));
> solve(abs(x)^2/3<1,x);

From this, we can conclude that the interval of convergence is $(-\sqrt{3},\sqrt{3})$.

Exercises

1. For the following functions and base points, determine what minimum order is required so that the Taylor polynomial approximates the function to within a tolerance of $0.1$ over the given interval.

   a)

   $f(x) = x\exp(x)$, base point $a=0$, interval $[-2,2]$.

   b)

   $$f(x) = \frac{3x}{x^2+1}$$, base point $a=0$, interval $[-0.9,0.9]$.

   c)

   $$f(x) = x^3\sin(x)$$, base point $a=0$, interval $[0 ,4\pi]$.

2. Use Maple to find the interval of convergence for the following series
   
   $$\sum_{n=1}^{\infty} \frac{(-1)^n}{5^n}(2x+3)^n$$
   
   

3. 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. 
      
      $$f(x)=\frac{1}{1-3x}$$
      
      

   2. 
      
      $$f(x)=3^x$$