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Subsections
The purpose of this lab is to learn how to determine convergence of series using Maple as well as introduce you to power series and their radius of convergence.
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_C18.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.
An infinite series is the sum of an infinite sequence. More precisely, the sum of an infinite series is defined as the limit of the sequence of the partial sums of the terms of the series , provided this limit exists. If no finite limit exists, then we say that the series is divergent.
The sum of an infinite series is defined as
, where is the partial sum of the first n terms of the series. However, because of the algebraic difficulty of expressing as a function of n, it is usually not possible to find sums by directly using the definition.
So, if we can generally not work from the definition, what can be done? There are several convergence tests that provide us with some needed tools. These are tests that tell us if a series converges, but in the case that the series does converge, does not tell us the sum of the series.
The series
diverges if
is nonzero or does not exist.
Consider the series
and
.
 If for , where is some integer and
converges, then
converges also.
 If for , where is some integer and
diverges, then
diverges also.
The Integral Test for convergence is a method used to test convergence of an infinite series of nonnegative terms.
The series
converges if and only if the integral
is finite, where is a positive, nonincreasing and continuous function defined on the interval and an for all .
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
as
.
Given the series
, suppose that
Then
 the series converges if ,
 the series diverges if ,
 the test is inconclusive if .
A power series about has the form
and a power series about has the form
where are the constant coefficients of the powers of .
The radius of convergence of a power series can usually be found by using the Tatio test:
Next, you would need to solve for the interval of values such that . Remember that power series always converge:
 If exists and is nonzero, the power series converges absolutely on some interval .
 If , the power series converges for all .
 If
, the power series converges only at
 Use any convergence test to determine if each of the following series converges or diverges and explain your answer.







 Use Maple to find the interval or radius of convergence for the each of the following series




Next: About this document ...
Up: lab_template
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Dina J. SolitroRassias
20180128