Math 1210-003
Introduction to Sequences and Limits
Spring 2012
1/17/12
Basic Sequences
DEFINITION: A sequence is an infinitely long, ordered set of numbers. The notation
is
used to describe an infinitely long sequence where
is the nth number in the sequence and is
often times described by some function of n. So
represents the ordered set of numbers
which continues indefinitely.
EXAMPLES: Given that
Here are some examples:
is a sequence we have a choice of how we can define each
.
describes the sequence
describes the sequence
describes the sequence
describes the sequence
describes the sequence
The Limit of a Sequence
In general there are two major types of end behaviors a sequence can exhibit.
1. Convergence: If a sequence,
, appears to get closer and closer to some finite number, L,
we say that the sequence converges to L. Furthermore, L is said to be the limit of the sequence
. The following are examples of converging sequences:
converges to 0
converges to 1
2. Divergence: If a sequence,
, does not converge, it is said to diverge. There are two
important ways in which a sequence can diverge. In the first case, a sequence can continue to
grow larger and larger (or more and more negative), in which case we say the sequence
diverges to (or
). In the second case, a sequence can alternate between two or more
Math 1210-003
Introduction to Sequences and Limits
Spring 2012
1/17/12
values indefinitely, in which case we say the sequence is alternating. Consider the following
examples:
converges to
converges to
alternating sequence
We are often interested in knowing when a sequence converges and when it diverges. If it
converges, we are further interested in knowing the limit of the sequence. Let's give a more
precise definition of a limit. [NOTE: If you don't understand the definition on the first pass,
that's ok, you won't be the first calculus student in the world to have trouble grasping this
concept. Give yourself time to absorb the idea and see it used in several examples or worded
differently].
DEFINITION: L is the limit of the sequence
exists a sufficiently large number N such that
if for any
, no matter how small, there
whenever
.
In other words, I give you a really small number of my choice, called , and you can always pick
a number, N, large enough so that after the Nth step in the sequence all of the numbers are less
than units away from the limit L.
In essence, as you proceed further along in the sequence, the numbers of the sequence
continue getting closer and closer to the limit, L. So if I were to ask the question, "At what point
in the sequence is the current number less than units apart from the limit, L?", you would
always be able to identify an appropriate place in the sequence.
The graph below illustrates a converging sequence.
ε
(
L
For the particular choice of in the sequence above, we see that every number in the sequence
after and 3rd number are all within units of the limit L. So for this value of we could set
and accurately say that
as long as
.
Math 1210-003
Introduction to Sequences and Limits
Spring 2012
1/17/12
Verifying the Limit of a Sequence
One can often take a reasonable guess at the limit of a sequence, but how can you be sure if
your guess is correct? To prove whether or not your guess is the true limit of the sequence, you
must start with the definition of a limit.
Let's suppose you have a sequence,
, and you take a guess at its limit, which we will refer
to as . It is now your task to show that given any
, you can find some number so that
whenever
, the inequality
always true. In other words, you must find a
systematic way of determining an appropriate value of given a value of . Normally, your
goal is to specify as a function of . That is,
.
Here are the steps you will typically follow to verify that L is the limit of the sequence
1. Substitute the definitions of
.
and L into the inequality
2. Simplify the inequality (removing the absolute value if at all possible) and solve for .
3. When step 2 is finished, your inequality should have the form
. Set
.
If the above steps are possible for the given sequence and the value of L, then L is the limit of
the sequence. Let's go through some concrete examples to help illustrate this procedure more
clearly.
EXAMPLES:
1. Let the sequence
be defined by
STEP 1 (substitute):
. Show that its limit is L = 0.
becomes
STEP 2 (simplify & solve):
Since
simplifies to
.
is always positive, then is always positive, so the absolute value is not
necessary here. Thus the inequality further simplifies to
Solving this inequality for
gives that
.
.
STEP 3 (define N): Let
Therefore the limit is correct since we have a systematic way of assigning N given any value of
. That is, all numbers in the sequence beyond the Nth number are within units of the limit,
, when
.
Math 1210-003
Introduction to Sequences and Limits
Spring 2012
1/17/12
For example, suppose we wish to know at what point in the sequence the numbers come within
units of the limit. We find that
. So all numbers in the
sequence beyond the 100,000th number are within
units of the limit,
.
2. Let the sequence
be defined by
STEP 1 (substitute):
. Show that its limit is L = 5.
becomes
STEP 2 (simplify & solve):
Since
simplifies to
is always positive, then
always equal to
is always positive, and so the expression
. Thus the inequality further simplifies to
Solving this inequality for
.
gives that
is
.
.
STEP 3 (define N): Let
Therefore the limit is correct since we have a systematic way of assigning N given any value of
. That is, all numbers in the sequence beyond the Nth number are within units of the limit,
, when
.
3. Let
be defined by
. Show that its limit is L = 2.
STEP 1 (substitute):
becomes
STEP 2 (simplify & solve):
Notice that the expression
simplifies to
.
simplifies to
.
is always equal to
. Thus the inequality further
Solving this inequality for
and simplify gives that
STEP 3 (define N): Let
Therefore the limit is correct since we have a systematic way of assigning N given any value of
. That is, all numbers in the sequence beyond the Nth number are within units of the limit,
, when
.
Math 1210-003
Introduction to Sequences and Limits
Spring 2012
1/17/12
More Complicated Examples
There are some interesting sequences for which the limiting values is no quite so obvious. For
example, when the sequence converges to an irrational number. Consider the following two
examples:
1. Banks have several different ways in which they can compound interest onto your balance.
Let represent the balance in your account at the beginning of the year, let represent the
interest rate (as a decimal), let represent the number of times per year that interest is
compounded onto your balance, and let
represent your accumulated balance at the end of
the year given that there were compoundings that year. You can calculated
by the
following formula:
It is clear how to calculate your balance if your balance is compounded a finite number of times
in the year (e.g. quarterly, monthly, daily, every second, etc.). What if it was compounded
continuously, as some banks do? This would be as if interest was compounded on your money
every instant, so essentially an infinite number of times per year. If we think of
as a
sequence, the balance at the end of the year for continuously compounded interest is
essentially the limit of the sequence
. As it turns out, the limit of this sequence is
.
Recall that e is an irrational number and
.
5.
The ancient Greeks had a clever way of approximating the
area of a circle using regular polygons. Consider the sequence of
regular polygons shown to the right. Assume that each polygon
has a radius of 1 (the radius of a regular polygon is the distance
from its center to one of its vertices). Let
be the sequence in
which is defined to be the area of the regular n-gon in this
sequence of regular polygons. Using some basic trigonometry, one
can find that
What shape are the polygons in this sequence approaching? They are becoming more and
more similar to the unit circle. It stands to reason that the sequence of polygon areas,
,
must be approaching the area of the unit circle, which is . In fact, the limit of the sequence,
is .