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.

- Functions , for example rational functions, that have vertical asymptotes in (or are not bounded on ).
- Integrals where the interval is unbounded, for example intervals like , , or .

We have already seen at least one example of the problems you can run
into if the function is unbounded. Recall the clearly absurd result

that is obtained by blindly applying the FTOC. The second type of problem, where the interval of integration is unbounded, occurs often in applications of calculus, such as the Laplace and Fourier transforms used to solve differential equations. It also occurs in testing certain kinds of infinite series for convergence or divergence, as we will learn later.

We start with the following definition.

- The interval of integration is unbounded.
- The function has an infinite discontinuity at some point in . That is, .

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 right-hand side exist. If

> ex1 := int(1/x,x=a..2);

> limit(ex1,a=0,right);The example above used the

> int(1/x,x=0..2);

The other two cases are handled similarly. You are asked to provide suitable definitions for them in one of the exercises.

> 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(1/x^2,x=2..infinity);

> a1[n]:=n->3-n^(-1); > seq(a1[n](i),i=1..10); > plot([[k,a1[n](k)]$k=1..10],style=point); > limit(a1[n](n),n=infinity);

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. The example below shows how series can be defined using the terms from above.

> S1:=m->sum(a1[n](j),j=1..m); > seq(S1[m](i),i=1..10); > plot([[k,S1[m](k)]$k=1..10],style=point); > limit(S1[n](n),n=infinity);

- Determine which of the regions described below have finite area. Use a limit for all improper integrals.
- A)
- The region below , above the axis, over the interval .
- B)
- The region below , above the axis, over the interval .

- Evaluate the following integrals in Maple by splitting the range of integration into separate ranges and completing separate integrals. Be sure to use limits for each integral.
- A)
- .
- B)
- .
- C)

- Define the sequence
as a function in Maple.
- List the first twenty terms.
- Plot the first twenty terms. Does it appear to converge or diverge? If it seems to converge, to what does it converge?
- Calculate the limit of the sequence and compare to your estimate.
- Write a function for the sum of the sequence and list the first twenty terms of the sequence of partial sums.
- Plot the at least the first twenty terms of the sequence of partial sums. Does it appear that the series converges or diverges? If it seems to converge, to what does it converge?
- Calculate a limit of the series to check your answers.

2017-10-24