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Sequences

A sequence is a function that computes an ordered list. For example, the average person in the United States uses 100 gallons of water each day. The function defined by (n) = 100n generates the terms of the sequence

100, 200, 300, 400, 500, 600, 700,…,

when n = 1, 2, 3, 4, 5, 6, 7, …. This function represents the gallons of water used by the average person after n days.

Sequences

A second example of a sequence involves investing money. If $100 is deposited into a savings account paying 5% interest compounded annually, then the function defined by g(n) = 100(1.05)n calculates the account balance after n years. The terms of the sequence are

g(1), g(2), g(3), g(4), g(5), g(6), g(7), …,

and can be approximated as

105, 110.25, 115.76, 121.55, 127.63, 134.01, 140.71, . . . .

A finite sequence is a function that has a set of natural numbers of the form

{1, 2, 3, …, n} as its domain. An infinite sequence has the set of natural

numbers as its domain.

Sequences

For example, the sequence of natural-number multiples of 2,

2, 4, 6, 8, 10, 12, 14, …, is infinite,

but the sequence of days in June,

1, 2, 3, 4, …, 29, 30, is finite.

Sequences

Instead of using f (x) notation to indicate a sequence, it is customary to use an, where an = (n).The letter n is used instead of x as a reminder that n represents a natural number. The elements in the range of a sequence, called the

terms of the sequence, are a1, a2, a3, …. The elements of both the domain and the range of a sequence are ordered. The first term is found by letting n = 1, the second term is found by letting n = 2, and so on. The general term, ornth

term, of the sequence is an.

Sequences

These figures show graphs of (x) = 2x and an = 2n. Notice that (x) is a continuous function, and an is discontinuous. To graph an,we plot points of the form (n, 2n) for n = 1, 2, 3,….

Example 1

Write the first five terms for each sequence.

a.

Solution

Replacing n in

with 1, 2, 3, 4, and 5 gives

Example 1

Write the first five terms for each sequence.

b.

Solution

Replace n in

with 1, 2, 3, 4, and 5 to obtain

Example 1

Write the first five terms for each sequence.

c.

Solution

Replacing n in

we have

Converge and Diverge

If the terms of an infinite sequence get closer and closer to some real number, the sequence is said to be convergent and to converge to that real number.

For example, the sequence defined by

approaches 0 as n becomes large.

Converge and Diverge

Thus an, is a convergent sequence that converges to 0. A graph of this sequence for

n = 1, 2, 3, …, 10 is shown here. The terms of an approach the horizontal axis.

Converge and Diverge

A sequence that does not converge to any number is divergent. The terms of the sequence are

1, 4, 9, 16, 25, 36, 49, 64, 81, ….

This sequence is divergent because as n becomes large, the values of do not approach a fixed number; rather, they increase without bound.

Recursive Definitions

Some sequences are defined by a recursive definition, one in which each term after the first term or first few terms is defined as an expression involving the previous term or terms. On the other hand, the sequences in Example 1 were defined explicitly, with a formula for an that does not depend on a previous term.

Example 2

Find the first four terms of each sequence.

a.

Solution

This is a recursive definition. We know a1 = 4. Since

an = 2 an – 1 +1,

Example 2

Find the first four terms of each sequence.

b.

Solution

This is a recursive definition. We know a1 = 2 and

an = an – 1 + n – 1.

MODELING INSECT POPULATION GROWTH

Example 3Frequently the population of a particular insect does not continue to grow indefinitely. Instead, its population grows rapidly at first, and then levels off because of competition for limited resources. In one study, the behavior of the winter moth was modeled with a sequence similar to the following, where an represents the population density in thousands per acre during year n.

MODELING INSECT POPULATION GROWTH

Example 3a. Give the table of values for n = 1, 2, 3, …, 10

Solution

Evaluate a1, a2, a3, …, a10.

and

MODELING INSECT POPULATION GROWTH

Example 3a. Give the table of values for n = 1, 2, 3, …, 10

Solution

Approximate values for n = 1, 2, 3, …, 10 are shown in the table.

MODELING INSECT POPULATION GROWTH

Example 3b. Graph the sequence. Describe what happens to the population density.

Solution

The graph of a sequence is a set of discrete points. Plot the points (1, 1), (2, 2.66), (3, 6.24), …,(10, 9.98), as shown here.

MODELING INSECT POPULATION GROWTH

Example 3Solution

At first, the insect population increases rapidly, and then oscillates about the line y = 9.7.

The oscillations become smaller as n increases, indicating that the population density may stabilize near 9.7 thousand per acre

Note In Example 3, the insect population stabilizes near the value

k = 9.7 thousand. This value of k can be found by solving the quadratic

equation k = 2.85k – .19k2. Why is this true?

Series and Summation Notation

Suppose a person has a starting

salary of $30,000 and receives a $2000 raise each year. Then,

30,000 32,000 34,000 36,000 38,000

are terms of the sequence that describe this person’s salaries over a 5-year

period.

Series and Summation Notation

The total earned is given by the finite series whose sum is $170,000. Any sequence can be used to define a series.

Take a look at the following slide to see what this looks like.

Series and Summation Notation

Any sequence can be used to define a series. For example, the infinite sequence

defines the terms of the infinite series

Series and Summation Notation

If a sequence has terms a1, a2, a3, …, then Sn is defined as the sum of the first n terms. That is,

The sum of the terms of a sequence, called a series, is written using summation notation. The symbol , the Greek capital letter sigma, is used to indicate a sum.

A finite series is an expression of the form

and an infinite series is an expression of the form

The letter iis called the index of summation.

Caution Do not confuse this use of i with the use of i to represent the imaginary unit. Other letters, such as k and j, may be used for the index of summation.

Example 4

Evaluate the series

Solution

Write each of the six terms, then evaluate the sum.

Example 4

Evaluate the series

Solution

Write each of the six terms, then evaluate the sum.

USING SUMMATION NOTATION WITH SUBSCRIPTS

Example 5Write the terms for each series. Evaluate each sum, if possible.

a.

Solution

USING SUMMATION NOTATION WITH SUBSCRIPTS

Example 5b.

Solution

Use the order of operations.

Substitute the given values for x1, x2, and x3.

If a1, a2, a3, …, an and b1, b2, b3, …, bn are two sequences, and c is a constant, then for every positive integer n,

(a)

(b)

(c)

(d)

Summation Properties

To prove Property (a), expand the series to obtain

where there are n terms of c, so the sum is nc.

Property (c) also can be proved by first expanding the series:

Commutative and associative properties.

Example 6

Use the summation properties to find each sum.

a.

Solution

Property (a) with n = 40 and c = 5.

Example 6

Use the summation properties to find each sum.

b.

Solution

Property (b) with c = 2 and ai = i

Summation rules

Simplify.

Example 6

c.

Solution

Property (d) with ai = 2i2 and bi = 3

Property (b) with c = 2 and ai = i2

Summation rules; Property (a)

Simplify.

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