Four Basic Questions for Infinite Sequences
1. What
is the nth
term of the sequence?
2. Can the terms of the sequence be given by a
formula?
3
. If the Series converges, what is the Limit?
4. If a sequence diverges is it bounded?
A. Explore the Definition of the Limit of a Sequence
Given the sequence defined by:
a) Use the trace command on the above graph to find
the coordinates of the
first point inside of the black band bounded by L -
e
and L + e
. You may need to modify the graph by zooming. This
value of n will serve for M in the definition. Why?
b) Change the value of e
above to e
= 0.01. Repeat part a), find the coordinates of the
first
point inside of the band. Determine a value for M.
c) Finish the table below.
d) Explain the definition of the limit in your own words.
B. The Four Basic Questions
For the calculus, one of the important properties
of infinite sequences is whether they converge to a
limit. In the previous exercise we saw that the terms
in the sequence seemed to approach a particular number.
The Online Encyclopedia of Integer Sequences at ATT
labs
http://www.research.att.com/~njas/sequences/index.html
The Formal Definition of the Limit of a Sequence
Let
L
be a real number. The limit of a sequence
{a
n }
is
L, written as
if for each
e > 0, there exists an
M > 0 such that
whenever n > M.
Finding the Limit of a Sequence Symbolically
Sometimes the symbolic tools in Mathcad are able
to determine the limit of a sequence. For example:
Given the sequence defined by the formula,
determine the limit of this sequence symbolically.
To complete our introduction to mathematical sequences,
we summarize as follows:
For values of k between 3.6 and 4 the behavior becomes
chaotic, in fact it may be used as a random number
generator!
Explore the Logistic Difference Equation
If k = 4, the logistic difference equation acts
like a random number generator. This is an example
of a bounded divergent sequence.
D. Make up your own sequence generating equations.
Cook up your own defining equations, and try to
determine the convergent or divergent propertiy of
your sequence.
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1. For each of the following sequences:
a) Graph the first 100 terms of the sequence.
b) List the first 15 terms of the sequence.
c) Find the limit symbolically.
Much attention has been given to the sequences
that are the discrete versions of the logistic differential
equation. The recursive equation is simple enough,
however the terms of the sequence may exhibit extremely
complex behavior. This equation is used as an introduction
to dynamical systems and chaos theory a hot area of
mathematical research.
The Logistic Difference Equation:
Formed by arranging pebbles in triangular patterns.
The first four triangular numbers are:
Defining Formulas and Algorithms
Sometimes it is possible to define a sequence of
numbers by an algebraic formula. For example, the nth
triangular number is given by:
While other times it may be possible to define a sequence
of numbers recursively as we did using Newton's Method
for finding the zeros of a function
. Recursively defined sequences require that we give
the algorithm an initial value.
A list of the first 5 terms:
An Introduction to Sequences
from Latin sequentia, literally, act of following
_______________________________________
1, 13, 17, 23, 38, 53, 5079, 57823
A number sequence is simply an ordered list of
numbers. The sequence may be finite or infinite, reveal
a pattern or appear entirely random. In this lab we
will investigate some of the properties of number sequences.
First we should note that in mathematical circles the
formal definition of a sequence is commonly cast in
the form of a function. For a sequence of numbers with
the first number of the sequence,
the second number and on, the natural function for
a sequence is:
The Domain is the positive integers
The Range is the terms of the sequence
The Formal Definition of a Sequence
A sequence is a function whose domain is the set
of positive integers.
Denoted by:
and also {
} for
The Earliest Studies of Number Sequences
While sequences of numbers appear on the earliest
of artifacts from prehistory, the study of sequences
as abstract patterns of numbers was most likely conducted
by the Pythagoreans circa 400 B.C.E. The Pythaogoreans
generated number sequences by geometrical arrangements
of pebbles, called figurate numbers. Although it was
a part of their belief that "Everthing is number",
it led to the birth of the branch of mathematics that
is today called number theory. Some examples of figurate
numbers include:
1. What is the smallest starting number that generates
a hailstone sequence with length at least 15?
2. Identify a starting number for which the hailstone
sequence has length at least 30. What is the length
of the sequence?
Since we have formally defined a sequence as a
function, you might be wondering if it is possible
to graph the sequence. It is possible, and the graph
looks like a scatterplot.
In the year 1202, Leonardo of Pisa published his
masterpiece, Liber
Abbaci
. The most celebrated problem in the text is the rabbit
problem, "How many pairs of rabbits can be bred
in one year from a single pair?" This sequence
is known today as the Fibonacci sequence and may be
calculated recursively as follows:
Generate a list of the first 15 terms of the Fibonacci
sequence using the computer.
1) Define the range variable n, and the recursive function.
2) Printout the sequence.
An interesting unsolved problem in mathematics
concerns what is called the "hailstone sequence."
This sequence is defined as follows: Start with any
positive integer. If that number is odd, then multiply
it by three and add one. If it is even, divide it by
two. Then repeat.
For example, starting with 5 we get the hailstone
sequence: 5, 16, 8, 4, 2, 1, 4, 2, 1, 4, 2, 1, ...
Here, the subsequence 4,2,1 is reached, resulting in
a loop. It is conjectured that no matter what number
you start with, you will always end up in the 4,2,1
loop. It has, in fact, been shown to hold for all starting
values up to 1,200,000,000,000. However, the conjecture
still has not been proven to hold for all numbers.
Exploration: Hailstone Sequences
The Length of the Hailstone sequence is the number of
terms in the sequence until the subsequence 4, 2, 1
appears.
