Mathematics 162                                Laboratory 7                          Week of March 8, 1993
	  
Name: _____________________________   Lab Partner: ___________________________
	  
Consulted with:  ____________________________________________________________

© Lafayette College, 1994.

Infinite Sequences and Series

Sequences

An infinite sequence is a function whose domain is the set of positive integers.

Part 0:  Examples

Consider the sequence

Some values of  f  are  f (1) = –1/2,  f (5) = 3/6 = 1/2, and  f (1004) = 1002/1005.  Notice that  f (–1),  f (2.5) and  f (e) make no sense since –1, 2.5, and e are not positive integers, and therefore not in the domain of  f.

Consider another sequence

Some values of this sequence are  f (2) = 4/2 = 2, and  f (6) = 64/720.

Notation:  We often denote a sequence by listing its range as an ordered set, since the positive integer corresponding to a given output can be determined from the output's position in the list.  For example, for the first sequence above, we might use any of the following notations:

In the first expression we list the first three terms (outputs of a sequence are called terms) and ask the reader to deduce the rule that generated them; this is not always easy to do, so this is not especially useful.  in the second expression we give the rule and the domain over which it is to be applied to generate the list of terms.  The third expression is the most concise, and therefore the one we will use most; it just give the rule, and the domain is implicit.  For the second sequence we would write

In this laboratory, you will use Mathematica's numerical and graphical tools to explore some infinite sequences, paying special attention to the limiting behavior of the sequences:  the patterns apparent in the terms corresponding to large inputs.

Part 1:  Listing and Plotting Terms of a Sequence

We can use the Table command to list some of the terms of a sequence.  (Of course, since sequences contain infinitely many terms, we cannot list all of them.)  Use

Table[{n, (n-2)/(n+1)}, {n,1,10}]

to list the first ten terms of the first example sequence: {(n – 2/(n + 1)}.  Now use

ListPlot[%]

to plot the pairs in your list.  (You may want to use the PlotStyle option to change the color of the plot.)  To see the behavior of the first fifty terms use

ListPlot[Table[{n, (n-2)/(n+1)}, {n,1,50}]]

What appears to be happening to the terms in the sequence as n gets larger?

Note:  We used ListPlot ( rather than Plot) to make these graphs because the domain ofthe sequence includes only the positive integers, not the numbers in between.  That is also why we didn't join the points of the graph with line segments.

Part 2:  Some More Examples of Sequences

In each of the following, use ListPlot to plot enough terms of each sequence to get an idea of its limiting behavior.  Write one or two sentences describing that behavior.

Part 3:  Convergent and Divergent Sequences

A sequence {an} is said to converge to the limit L (which must be a number) if we can make an as close to L as we want by making n large enough.  Which of the sequences in Part 2 converge?  In each case, give the limit.

A sequence diverge if it does not converge to a real number.  Briefly describe two ways a sequence can diverge.  Illustrate each of these ways with an example from Part 2.

Define a sequence of your own and decide whether it converges or diverges.  If it converges, give its limit; if it diverges explain what sort of divergence occurs. (Is it one of the two types you described above, or some new way of diverging?)  Explain how you came to your conclusion.

Series

An infinite series is defined as the sum of all the terms of an infinite sequence.  We write the series with terms {an} as

Part 4:  Sequences of partial sums

The way we make sense out of infinite series (a sum that never ends) is by defining a new sequence, the sequence of partial sums.  the kth partial sum is defined by

We find the sum of the infinite series by finding the limit of the sequence of partial sums.

We can compute the partial sum sk using a Mathematica command of the form Sum[a[n], {n,1,k}].  This always give exact values.  To get numeric approximations we can use either the N[ ] command, or the numeric sum command, NSum[a[n],{n,1,k}].

For each of the series below, determine if the sequence of partial sums for the infinite series below converges.  Explain why you think it converges or diverges, and estimate the limit if it converges.

(a)

(b)

(c)

(d)