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Subsections
The purpose of this lab is to use Maple to introduce you to the notion
of improper integral and to give you practice with this concept by
using it to prove convergence or divergence of integrals involving
unbounded integrands or unbounded intervals or both. You will also be
introduced to to the concepts of convergence and divergence of sequences
and series.
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\Improper_int_start_A14.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.
We start with the following definition.
To see how to handle the problem of an unbounded integrand, we start
with the following special cases.
Definition 2
Suppose that is continuous on , but
. Then we define
provided that the limit on the righthand side exists and is finite,
in which case we say the integral converges and is equal to the value
of the limit. If the limit is infinite or doesn't exist, we say the
integral diverges or fails to exist and we cannot compute it.
Definition 3
Suppose that is continuous on , but
. Then we define
provided that the limit on the righthand side exists and is finite,
in which case we say the integral converges and is equal to the value
of the limit. If the limit is infinite or doesn't exist, we say the
integral diverges or fails to exist and we cannot compute it.
Cases where has an infinite discontinuity only at an interior
point are handled by writing
and using the definitions to see if the integrals on the righthand
side exist. If both exist then the integral on the lefthand
side exists. If either of the integrals on the righthand side
diverges, then
does not exist.
Here is a simple example using Maple to show that
doesn't exist.
> ex1 := int(1/x,x=a..2);
> limit(ex1,a=0,right);
The example above used the right option to limit
because the righthand limit was needed. If you need a lefthand
limit, use the left option in the limit command. Maple can usually do the limit within the int command.
> int(1/x,x=0..2);
These are handled in a similar fashion by using limits. The definition
we need the most is given below.
Definition 4
Suppose is continuous on the unbounded interval .
Then we define
provided the limit on the righthand side exists and is finite, in
which case we say the integral converges and and is equal to the value
of the limit. If the limit is infinite or fails to exist we say the
integral diverges or fails to exist.
The other two cases are handled similarly. You are asked to provide
suitable definitions for them in one of the exercises.
Using the definition for
.
> ex2:=int(1/x^2,x=2..a);
> limit(ex2,a=infinity);
This command shows that Maple takes the limit definition into account in the int command.
> int(1/x^2,x=2..infinity);
An infinite sequence is a list of numbers defined by the formula where is the set of positive integers . A sequence is said to converge if the numbers in the in the list approach a single value as approaches . If the numbers in the list increase or decrease without bound as gets larger, then we say that the sequence diverges.
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 not always possible to find sums by directly using the definition.
 The gamma function is an example of an improper integral often used to approximate noninteger factorials and is defined below:
Evaluate by calculating the improper integral for
and for each integer value of , check your answer by calculating .
 Both of the following improper integrals given below do not exist. Show, by calculating a limit, why they do not exist.
 A

 B

 Plot on the interval
. Recall from Calculus II that the volume of a solid of revolution formed by rotating about the axis over the interval is
and the surface area is
Find the volume and surface area of the solid obtained by revolving the curve
about the axis, between and .
 Write the sequence
as a function. List the first fifteen terms.
 Plot the first fifteen terms. Does it appear to converge or diverge? If it seems to converge, to what does it converge?
 Calculate the limit of the sequence as approaches . Is your answer close to your approximation above?
 Write a function for the sum of the sequence. List the first ten terms of the sequence of partial sums.
 Plot the at least the first ten terms of the sequence of partial sums (You may want to restrict the range to
). Does it appear that the series converges or diverges? If it seems to converge, to what does it converge?
 Calculate the limit of the series as approaches .
Next: About this document ...
Up: lab_template
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Dina J. SolitroRassias
20140929