Chapter 9:   Sequences and Series

Section 9.1: Sequences and Summation Notation (page 622)

Function Notation: Normally if the name of a function is a and n is an element of the domain of the function, we use a(n) to denote the unique range element which is associated with the domain element n by the function a. However, historical useage dictates a slightly different functional notation for sequences.

If a is the name of a sequence and n is a domain element then we use a_n to denote the unique range element which is associated with the domain element n by the function a.

The definition of sequence is pretty simply stated, but there are many special consequences of the fact that the domain of a sequence is N.  Sequences have been studied for centuries.  Sequences have been studied with and without the concept of function.  Quite a bit of special (mostly historical in origin) terminology and notation is used in a discussion of sequences.

When working with sequences, range elements are frequently called terms of the sequence. For example:

• The range element associated with the domain element 1 is called the first term of the sequence.
• The range element associated with the domain element 6 is called the sixth term of the sequence.
• The range element associated with the domain element n is called the nth term of the sequence.

• Because the domain of every sequence is N, the graph of a sequence will consist of a set of discrete points.
• Because the domain of every sequence is N, it is possible to speak of the first domain element.
• Because the domain of every sequence is N, it is possible to speak of the next domain element.
• Because the domain of every sequence is N, it is not possible to speak of the last domain element.
• Because the domain of every sequence is N, it is possible to ask about and compute the sum of the first k terms. This is usually called the kth partial sum of the sequence.

Definition:Summation Notation: The sum of the first n terms of a sequence named a is represented by
i is the index of summation, 1 is the lower limit and n is the upper limit.

Definition: The sum of all the terms of a sequence is called an infinite series

Notation:The n^th partial sum of a sequence named a is symbolized by and the infinite series associated with the sequence named a is symbolized by

Section 9.2: Arithmetic Sequences (page 635)

Definition: An arithmetic sequence is a sequence whose consecutive terms have a common difference.

Definition: The difference between consecutive terms of an arithmetic sequence is called the common difference of the sequence.

Equivalent Definition: An arithmetic sequence f is a function whose rule may be expressed as a linear equation of the form f(n) = dn + b where d is the common difference and b is the difference f(1) – d.

Fact: The n^th term of an arithmetic sequence named a is given by the rule a_n = dn + c where d is the common difference and c is a constant.
Observe that if we used normal functional notation the rule would be written as
a(n) = dn + c
which looks like the familiar linear function whose rule is usually presented as
f(x) = mx + b.

Comments:Compare the equivalent definition of an arithmetic sequence with the definition of a linear function to conclude the following.

The domain of a linear function is R
The domain of an arithmetic sequence is N

The rule for a linear function is f(x) = mx + b
The rule for an arithmetic sequence is a(x) = dx + b

The number b in the rule for an arithmetic sequence is the range value associated with 0 (if there were such a range element) so it corresponds exactly to the y-intercept of the linear function.

The common difference d is nothing more than the slope as you move from one range element to the next.

The slope of the line joining two terms (x, f(x)) and (x + 1, f(x + 1)) of the sequence f is given by.

A casual approach is to view an arithmetic sequence as a linear function with domain N.

Comment: Given any two pieces of information about an arithmetic sequence it is possible to determine its rule.  The next three problem types illustrate the point.

Problem Type 1: If you are given the common difference and the first term of the arithmetic sequence, then it is possible to write the rule for the function. This is comparable to the slope-intercept situation/problem when working with linear functions.

Example: Suppose an arithmetic sequence named h has a common difference 8 and the first term is -5. Find the rule for the function h.
Solution: Since the function is an arithmetic sequence its rule is of the form
h(n) = dn + b. In our case the common difference d is 8, so the rule for h has the form
h(n) =8n + b. Because the first term is -5, b = -5 – 8 = -13 and the rule for the desired arithmetic sequence is given by h(n) = 8n – 13.

Problem Type 2: If you are given the common difference d and one term of an arithmetic sequence, then it is possible to write the rule for the function. This is comparable to the point-slope situation/problem when working with linear functions.

Example: Suppose an arithmetic sequence named h has a common difference 3 and the fifth term is 12. Find the rule for the function h.
Solution: Since the function is an arithmetic sequence its rule is of the form
h(n) = dn + b. In our case the common difference d is 3, so the rule for h has the form
h(n) =3n + b. Because the fifth term is 12, h(5) = 12, but according to the partially determined rule h(5) = 3(5) + b = 15 + b. These two representations for h(5) yield the equation 12 = 15 + b. Clearly then b = -3 and the rule for the desired arithmetic sequence is given by
h(n) = 3n – 3.

Problem Type 3: If you are given two terms of an arithmetic sequence, then it is possible to write the rule for the function. This is comparable to the two point situation/problem when working with linear functions.

Example: Suppose the fourth term an arithmetic sequence named k is 10 and the seventh term is 28. Find the rule for the function h.

Solution: Since the function is an arithmetic sequence its rule is of the form
h(n) = dn + b. The difference between the seventh and fourth terms is 3d and is also equal to 28 – 10 = 18. That means 3d = 18 and so the common difference d is 6.

The rule for h has the form h(n) =6n + b. Because the fourth term is 10, h(4) = 10, but according to the partially determined rule h(4) = 6(4) + b = 24 + b. These two representations for h(4) yield the equation 10 = 24 + b. Clearly then b = -14 and the rule for the desired arithmetic sequence is given by h(n) = 6n – 14.

Comment: The formula for the nth partial sum of an arithmetic sequence named a is:

Section 9.3: Geometric Sequences (page 640)

Definition:A sequence a is a geometric sequence if the ratio of consecutive terms of a is the same for all terms of the sequence.

Definition: The ratio between consecutive terms of a geometric sequence is called the common ratio of the sequence.

Fact: The n^th term of a geometric sequence named a is given by the rule

a_n = a₁r^{n - 1}

where r is the common ratio.

Observe that if we used normal functional notation the rule would be written as

a(n) = a₁r^{n - 1}

which looks like a familiar exponential function whose rule is normally presented as
f(x) = ar^x.

Section 9.5: Mathematical Induction

Section 9.6: The Binomial Theorem

Good Explanation