Sequences and Series

The bottom line, a sequence is an ordered list or an ordered set of numbers. The idea of ordering is important!

We call the individual objects in the list terms. We count the terms in the listing from left to right. This need not be in numerical order, but order must not be changed. We often wrap the listing in braces { } like normal sets, but keep in mind that ordering matters for sequences, but not for sets in general.

Examples:

1. 0, 1, 2, 3, 4 is a sequence with four terms. The third term is the number 2.
2. 2, 0, 1, 3, 4 is a different sequence with four terms. The third term is the number 1.
3. 1, 3, 5, 7, 9 is the sequence of odd digits. The fifth term is the number 9.
4. 1, 5, 9, 13, . . . , 33, 37 is a sequence with 13 terms.

   The dots ( . . . ) in the middle indicates that the pattern continues until the next number appears.

   Identifying the pattern is the key to working with sequences. What do we do to get each successive term?

   What is the 7th term?

5. 1, 4, 9, 16, 25, . . . , 100 is a sequence with ten terms.

   What is the pattern?

   What is the 7th term?

6. 0, -1, 2, -3, 4, -5 is an alternating sequence with six terms. The signs alternate from term to term.

Sequences can be finite like the ones above or they can be infinite like 10, 11, 12, 13, 14, . . . The three dots ( . . . ) at the end say the pattern continues forever.

Examples:

1. 0, 1, 2, 3, 4, 5, . . . is a infinite sequence, sometimes called the sequence of whole numbers. Its 10th term is the number 9. (Really, count it out.)
2. 1, 2, 3, 4, 5, . . . is also an infinite sequence, sometimes called the sequence of natural numbers.
3. -1, 2, -3, 4, -5, . . . is also an alternating infinite sequence

Warning: We are not allowed to rearrange any infinite sequence for any purpose! Remember ordering is important.

A curious fact about infinite sets. Analyzing their sizes can lead to interesting results. Sets 1 and 2 above are exactly the same size! No way, you say? Set 2 has the number zero and Set 1 does not. Well, think about the function . The x values come from the the sequence 0, 1, 2, 3, 4, 5, . . . and the y values are the sequence 1, 2, 3, 4, 5, . . It is easy to see that we can match up every x with exactly one y. This is called a one-to-one correspondence. Finding it is enough to prove the sizes are the same. This also shows that a sequence is nothing more than the range of some function with the specific domain of the counting or natural numbers.

Calculating specific terms leads to an "n^th term formula."

Before you can create a rule of calculation, you need to realize that sequences are functions with the specific domain of the counting numbers {1, 2, 3, 4, 5, ...}. So the n replaces x as the input variable and instead of writing y, we use a_n as the output variable.

Notice for the sequences in part 2 above, each is just a counting process. So, we can easily write a rule to calculate any term desired.

Examples:

1. For {0, 1, 2, 3, 4} and {0, 1, 2, 3, 4, 5, . . .} , the formula is . But this is unhandy since we need the previous term. It is easier to notice that . This is the n^th term formula.

   Where's the difference in the two sequences? In the domain of the sequences. An infinite sequence is understood to have the domain of the natural or counting numbers. (Sometimes the whole numbers are used when it is convenient.) What we are doing is setting up a one-to-one correspondence between the set of natural numbers and an ordered list of values (the terms).

2. For {10, 11, 12, 13, 14, . . .}, the formula is .
3. For {1, 4, 9, 16, 25, . . . , 100}, the formula is .
4. The following sequence is very different: {1, 1, 2, 3, 5, 8, 13, ...}. This is the Fibonacci sequence. In this case we are forced to write a formula that explains the next term in terms of the previous two terms.
   • Before I go further, see if you can analyze the process creating the terms.
   • Now see if you can complete the formula. This one has to set the first two terms specifically, so a₁ = 1 and a₂ = 1. Then a_n =

 

More About Notations In function notation, to ask for a value at 1 in a discrete function, we would write f (1). In sequence notation we write a1. The two ways of asking for a value produce exactly the same result when we are thinking of sequences such as the cost of the fifth item of production. Before we would say C(5). Now we can also say C5.

Series and Summation Notation

It is often valuable to talk about adding the terms of a sequence. This creates a series. We also introduce a notation to show that we want to add up the terms, called sigma notation. The process is demonstrated below.

Examples:

1. For {0, 1, 2, 3, 4}, with the formula

   The finite series is 0 + 1 + 2 + 3 + 4.

   In summation notation we write .

   The letter i is called the index. It has no relationship to the complex symbol i. We depend on context to keep this clear.

2. For {0, 1, 2, 3, 4, 5, . . .} , the formula is .

   The finite series is 0 + 1 + 2 + 3 + 4 + . . .

   In summation notation we write

   We can stick with the letter n for the index since the upper end to the summation is infinity.

3. For the alternating infinite sequence -1, 2, -3, 4, -5, . . ., the formula is

   The finite series is -1 + 2 - 3 + 4 -. . .

   In summation notation we write

   Notice how we used the powers of -1 to alternate the sign.

4. For the alternating infinite sequence -1, 2, -4, 8, -16, 32, . . ., the formula is

The infinite series is -1 + 2 - 3 + 4 -. . .

In summation notation we write

Notice again how we used the powers of -1 to alternate the sign.

The next step is to decide when we can write convenient formulas to actually calculate the n^th term and the actual sum of series when it is useful.

In your next lesson we'll discuss arithmetic sequences and series and how to apply them.