DEFINTION:

Usually n ₀ is 1, but this not always true. A sequence can start anywhere. The most important trait of a sequence is that it has a specific ordering.

EXAMPLE 1:

2, 4, 6, 8, 10, . . . and 4, 2, 8, 6, 12, 10, . . . are two different sequences.

As the definition states, a sequence is a function. This means that a sequence can be denoted in function notation such as in the following example.

A sequence can be defined recursively by giving the value(s) of the initial term or terms, and a recursion formula for calculating any later term from terms that precede it.

EXAMPLE 2:

F₁ = 1, F₂ = 1, F_{n + 1} = F_n + F_{n - 1}

1, 1, 2, 3, 5, 8, 13, . . .

This is the Fibonacci sequence.

Here are some more examples of sequences.

EXAMPLE 3:

Here are the first few terms of this sequence.

EXAMPLE 4:

Here are the first few terms of this sequence.

EXAMPLE 5:

Here are the first few terms of this sequence.

Now that I have shown you a few examples of what a sequence is, let us start to develop the formula for a sequence given a list of terms.

EXAMPLE 6:

Find the sequence's formula for 1, -4, 9, -16, 25, . . .

SOLUTION:

Notice that the terms are the squares of the counting numbers, but they are also alternating in sign, so we must come up with the formula for the nth term that allows for this trait. To take care of the alternating sign, I will use (-1) ^{n + 1}, so the formula for the nth term of this sequence is the following.

a _n = (-1) ^{n + 1} n ²

EXAMPLE 7:

Find the sequence's formula for 2, 0, 2, 0, 2, 0, . . .

SOLUTION:

It looks like half of the time the term of the sequence will be 0 and the other half 2. We know that 1 + 1 = 2 and 1 - 1 = 0, therefore the nth term formula is a _n = 1 + (-1) ^{n + 1}.

One of the most important topics that we will be concerned with when looking at sequences is whether they converge or diverge. If a sequence approaches a unique limiting value L, then we can say that the sequence converges. Before we can go any farther on this topic, we must talk about the concept of a subsequence.

FACT:

If the terms of one sequence appear in another sequence in their given order, we will call the first sequence a subsequence of the second.

EXAMPLE 8:

2, 4, 6, 8, . . . 2n, . . . is a subsequence of even integers.

EXAMPLE 9:

4, 8, 12, 16, 20, . . . 4n, . . . is a subsequence of the multiples of 4.

Why are we concerned with subsequences? There are two reasons why we are. The first reason states that if {a _n} converges to L, then all of its subsequences will converge to L. The second reason states that if any subsequence of a sequence {a _n} diverges or if any two subsequences have different limits, then {a _n} diverges. The second reason is important in determining the convergence or divergence of a sequence.

EXAMPLE 10:

Determine if a _n = 1 + (-1) ^{n + 1} converges or diverges.

SOLUTION:

One subsequence of this sequence is 2, 2, 2, 2, . . . which converges to 2.

Another subsequence of this sequence is 0, 0, 0, 0, . . . which converges to 0.

Therefore, by the fact that we have two different subsequences of the same sequence going to different limits, the original sequence diverges.

EXAMPLE 11:

Determine if a _n = cos (n p ) converges or diverges.

SOLUTION:

Let us list some terms of this sequence.

a ₁ = cos p = -1, a ₂ = cos 2p = 1, a ₃ = cos 3p = -1, a ₄ = cos 4p = 1

One subsequence of this sequence will be 1, 1, 1, 1, . . . which converges to 1.

Another subsequence will be -1, -1, -1, -1, . . . which converges to -1.

Since the two different subsequences of the same sequence are going to different limits, then the sequence does not converge.

Another type of a subsequence is the tail of the sequence.

DEFINITION:

A tail of a sequence is a subsequence that consists of all terms of the sequence from some index N on.

Now let us discuss some traits a sequence can have.

Consider the sequence 1, 2, 3, 4, . . . n, . . .. Notice that this sequence is continually growing. In fact, a ₁ < a ₂ < a ₃ < . . . < a _n < . . ..

DEFINITION:

A sequence {a _n} with the property a _n £ a _{n + 1} for all n is called a non-decreasing sequence.

EXAMPLE 12:

Here is another example of a non-decreasing sequence.

Here is the plot of the terms of this sequence.

FACT:

A constant sequence is also non-decreasing.

So here is the first trait a sequence can have. Another trait is that the sequence is bounded from above. What this means is that there is some number, M such that a _n £ M for all n.

DEFINITION:

A sequence {a _n} is bounded from above it there exists a number M such that a _n £ M for all n. The number M is an upper bound for {a _n}. If M is an upper bound for {a _n}, but no number less than M is an upper bound for {a _n}, then M is the least upper bound for {a _n}.

EXAMPLE 13:

1, 2, 3, 4, 5, . . . n, . . . has no upper bound. Here is the plot of the first 50 terms of this sequence to prove it.

EXAMPLE 14:

Here is the plot of the first 50 terms and the upper bound to prove it.

FACT:

A non-decreasing sequence that is bounded from above always has a least upper bound.

Why should we be concerned with the least upper bound? Remember that our goal is to determine if a sequence converges or diverges, so if a non decreasing sequence is bounded from above, the it will converge and vice versa.

FACT:

A non-decreasing sequence of real numbers converges if and only if it is bounded from above.

FACT:

If a non-decreasing sequence converges, then it converges to its least upper bound.

EXAMPLE 15:

Determine if the following sequence is non-decreasing and bounded from above.

SOLUTION:

Here are the first few terms of this sequence.

This does not give us a true picture on how this sequence behaves, so let us a computer algebra system to generate a list of the first 20 terms.

Here is the list of the first 20 terms.

{3, 20, 210, 3024, 55440, 1235520, 32432400, 980179200, 33522128640, 1279935820800, 53970627110400, 2490952020480000, 124903451312640000, 6761440164390912000, 393008709555221760000, 24412776311194951680000, 1613955767240110694400000, 113146793787569865523200, 838417741965827035269120000, 654764331820982873230540800000}

As you can see by this list, it is a non-decreasing sequence, but it is not bounded from above. Therefore, it does not converge.

EXAMPLE 16:

Determine if the following sequence is non-decreasing and bounded from above.

SOLUTION:

Here is the first four terms of this sequence.

After the third term, the sequence starts to decrease, so it is not non-decreasing. But, 4.5 bound it from above. Here is the plot of this sequence with its bound.

Do you think that this sequence converges or diverges? I would say that it converges, because n! will eventually grow faster than 3 ⁿ.

 EXAMPLE 17:

Does the following sequence converge or diverge?

 SOLUTION:

Again, let us look at a few terms of this sequence.

If you plug these expressions into your calculator, you will see that at first the sequence is non-decreasing, but then it start to decrease. Therefore, it is not non-decreasing. So, how are we going to determine whether or not this sequence converges or diverges? Let us look at a plot of this sequence. I will start this plot at 1st term. Here is the plot.

As you should see, as n approaches infinity, the sequence a _n is approaching 0. Therefore, this sequence converges.

EXAMPLE 18:

Does the following sequence converge or diverge?

SOLUTION:

The easiest way to determine the convergence or divergence of this sequence is to look at the plot of the terms of the sequence. Here is the plot.

 As you can see, at first n ⁴¹ is growing faster than 19 ⁿ, but eventually 19 ⁿ takes over. From the plot, you can conclude that the sequence converges to zero.