Sequences

A sequence is an infinite list of numbers in a particular order.

A specific number in a sequence is called a term.

Terms are named in chronological order. (ex: first term, second term, third term, etc.)

Terms are expressed in the following notation:

a1 = first term

a2 = second term

a3 = third term

an = nth term

Sequences may be expressed in two ways.

1. {an}n=k, where k is the starting index

2. an = f(n)

To write the sequence, we need to know its terms. Let’s start with the first term. To find a1, plug the starting index n=1 into an

a1 = 2(1)+1 = 2+1 = 3

Use the same method to find a2, a3, a4, and so on

a2 = 2(2)+1 = 4+1 = 5

a3 = 2(3)+1 = 6+1 = 7

a4 = 2(4) + 1 = 8+1 = 9

Now that we know the values for a1, a2, a3, and a4, we can write them into the sequence

{3, 5, 7, 9, ... }

Example Problem 2.1.2 Answer

The first step is to find the common difference.

d = an - an-1
d = a2 - a2-1 = a2 - a1
d = 10 - 3
d = 7

Next, plug d into the arithmetic sequence formula

an = dn + c
an = 7n + c

To find the value of the constant, choose a term to plug into the formula. In this example the first term will be used; however, any term may be used.

a1 = 7(1) + c
3 = 7 + c
c = -4

Now knowing the values of d and c, plug them back into the arithmetic sequence formula, to find that

an = 7n - 4

To graph the terms of a sequence, plug the values into the coordinates

(n, an)

Looking at the first ten, the terms appear to be decreasing, at first dramatically and then leveling off.

Looking at the first fifty and hundred, the graph appears to have leveled off at y = 1. As n gets larger and larger, the terms seem to get closer to 1.

The value 1 is known as the limit of the sequence as n approaches infinity. You may also hear that the sequence is bounded to 1, or that 1 is the lower bound.

If there exists a limit for a sequence, it is written as

lim an = L

n->∞

And in the case of example 2.1.5, the limit is written as

= 1

Tip: You can plug the formula for an into one of the Y equations, substituting n for x, on your graphing calculator. Going to the Table screen, you can find any nth term by plugging in ‘x=n’. The corresponding term an will be in the Y column. The Table screen is a great way to see the hundredth, thousandth, or any larger term.

Sequence Properties

1. If the next term of a sequence is either always greater than or always less than the previous term, then the sequence is known as monotonic.
2. If the next term of a sequence is always greater than the previous term, or if an > an-1, then the sequence is known as monotonically increasing.
3. If the next term of a sequence is always less than the previous term, or if an < an-1, then the sequence is known as monotonically decreasing.
   
4. If the term an gets closer to a value L as n approaches infinity, then lim an = L.
5. If a sequence increases without bound, then the lim an = ∞.
6. If a sequence decreases without bound, then the lim an = -∞.
7. If an = f(n) and there exists a function f(x) such that f(n)=f(x), and the lim f(x) = L, then the lim an = L.
8. Given an ≤ bn ≤ cn, and lim an = L = lim cn, then lim bn = L also.

Properties of Limits

We want to see if exists.

With sequences, more than likely you will have to change the formula in order to check if it converges or not, and this one is no exception.

Using the property of combining exponents with the same base,

=

Since the exponent on the second 4 is negative, you can bring it up to get

Following

,

Let’s start with finding the first four terms.

a1 =

a2 =

a3 =

a4 =

Looking at the first four terms, the terms are decreasing as n becomes larger. Also, you can see that the numbers in the numerator are always canceled out by numbers in the denominator, leaving the product of two numbers in the denominator. With algebraic manipulation, you can see that the term

an =

With the new format of an, you can find the limit of to see

= 0.

Therefore, the sequence converges to 0.

Let’s find the first five terms.

a1 =

a2 =

a3 =

a4 =

a5 =

Looking at the first five terms, you can see that the first term is equal to the fifth term. Because of the behavior of the sine function, the sequence will forever cycle through the values {1, 0, -1, 0, 1, 0, ...}. Since there is no value to which the sequence heads to as n approaches infinity,

= DNE

And since the limit does not exist, the sequence diverges.

This example will be approached by first looking at the numerator of the term.

Since this is a cosine function, you know the value will never be less than -1 or be greater than 1. As such,

-1 ≤ ≤ 1

Now since is being divided by n in the sequence, the next step will be to divide by n across the inequality.

The next step is to take the limit as n approaches infinity of the left and right sides.

= 0 =

Since they both head toward the same limit, this means that, according to the Squeeze Theorem,

= 0

Therefore, the sequence converges to 0.

We are looking to see if exists.

When n approaches infinity, the value of the sequence yields

.

Since it meets one of the conditions for L'Hôpital’s Rule, we can take the derivative of the top and the bottom and check the limit of that.

Taking the derivative, we are now looking to see if exists.

Again, as n approaches infinity, the value of the limit is

.

So we do L'Hôpital’s Rule again.

Now we are looking to see if exists, and it does.

= 0.

This tells us that the sequence n=1 converges to 0.