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Infinite Sequences and Summation Notation
SECTION 11.1
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Infinite Sequences
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Infinite Sequences
The ordered list of numbers 2, 4, 8, 16, 32,…is called an
infinite sequence. The list is ordered simply because order
makes a difference.
The sequence 2, 8, 4, 16, 32,… contains the same
numbers but in a different order. Therefore, it is a different
infinite sequence.
An infinite sequence can be thought of as a pairing
between positive integers and real numbers.
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Infinite Sequences
For example, 1, 4, 9, 16, 25, 36,…, n2,… pairs a positive integer with its square.
This pairing of numbers enables us to define an infinite sequence as a function whose domain is the positive integers.
Definition of an Infinite Sequence
An infinite sequence is a function whose domain is the positive integers and whose range is a set of real numbers.
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Infinite Sequences
Although the positive integers do not include zero, it is occasionally convenient to include zero in the domain of an infinite sequence. Also, we will frequently use the word sequence instead of the phrase infinite sequence.
As an example of a sequence, let f (n) = 2n – 1. The range of this function is
f (1), f (2), f (3), f (4), …, f (n), …
1, 3, 5, 7, …, 2n – 1, …
The elements in the range of a sequence are called the terms of the sequence.
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Infinite Sequences
For our example, the terms are 1, 3, 5, 7,…, 2n – 1,…. The
first term of the sequence is 1, the second term is 3, and
so on. The nth term, or the general term, is 2n – 1.
Rather than use functional notation for sequences, it is customary to use subscript notation. Thus an represents the nth term of a sequence. Using this notation, we would write
an = 2n – 1
Thus a1 = 1, a2 = 3, a3 = 5, a4 = 7, and so on.
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Example 1 – Find the Terms of a Sequence
a. Find the first three terms of the sequence .
b. Find the eighth term of the sequence .
Solution:
a.
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Example 1 – Solution
b.
cont’d
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Infinite Sequences
An alternating sequence is one in which the signs of the terms alternate between positive and negative.
A recursively-defined sequence is one in which each succeeding term of the sequence is defined by using some of the preceding terms. For example, let a1 = 1, a2 = 1, and an + 1 = an – 1 + an.
a3 = a1 + a2 = 1 + 1 = 2
a4 = a2 + a3 = 1 + 2 = 3
a5 = a3 + a4 = 2 + 3 = 5
a6 = a4 + a5 = 3 + 5 = 8
This recursively defined sequence 1, 1, 2, 3, 5, 8,… is called the Fibonacci sequence.
n = 2
n = 3
n = 4
n = 5
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Factorials
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Factorials
It is possible to find an nth-term formula for the sequence defined recursively using a1 = 1 and an = nan - 1. Consider the term a5 of that sequence.
a5 = 5a4
= 5 4a3
= 5 4 3a2
= 5 4 3 2a1
= 5 4 3 2 1
a4 = 4a3
a3 = 3a2
a2 = 2a1
a1 = 1
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Factorials
Continuing in this manner for an, we have
an = nan – 1
= n(n – 1)an – 2
= n(n – 1)(n – 2)an – 3
= n(n – 1)(n – 2)(n – 3) ... 2 1
The number n (n – 1) … 3 2 1 is called n factorial and is written n!.
…
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Factorials
Definition of the Factorial of a Number
If n is a positive integer, then n!, which is read “n factorial,” is
n! = n (n – 1) … 3 2 1
We also define
0! = 1
Example
5! = 5 4 3 2 1 = 120
10! = 10 9 8 7 6 5 4 3 2 1 = 3,628,800
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Factorials
Note that we can write 12! as
12! = 12 11! = 12 11 10! = 12 11 10 9!
In general,
n! = n (n – 1)!
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Example 4 – Evaluate Factorial Expressions
Evaluate each factorial expression.
a. b. 6! – 4!
Solution:
a.
b. 6! – 4! = (6 5 4 3 2 1) – (4 3 2 1)
= 720 – 24
= 696
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Partial Sums and Summation Notation
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Partial Sums and Summation Notation
Another important way of obtaining a sequence is by adding the terms of a given sequence.
For example, consider the sequence whose general term is given by . The terms of this sequence are
From this sequence we can generate a new sequence that is the sum of the terms of .
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Partial Sums and Summation Notation
and, in general,
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Partial Sums and Summation Notation
The term Sn is called the nth partial sum of the infinite sequence, and the sequence S1, S2, S3, …, Sn… is called the sequence of partial sums.
A convenient notation used for partial sums is called summation notation. The sum of the first n terms of a sequence an is represented by using the Greek letter (sigma).
ai = a1 + a2 + a3 + … + an
This sum is called a series.
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Partial Sums and Summation Notation
It is a finite series because it is the sum of a finite number of terms of the sequence. The letter i is called the index of the summation; n is the upper limit of the summation; 1 is the lower limit of the summation.
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Example 5 – Evaluate a Series
a. Let . Evaluate .
b. Let an = (–1)nn2. Evaluate .
Solution:
a.
b.
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Partial Sums and Summation Notation
Properties of Summation Notation
If an and bn are sequences and c is a real number, then
1.
2.
3.
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Arithmetic Sequences and Series
SECTION 11.2
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Arithmetic Sequences
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Arithmetic Sequences
Note that in the sequence
2, 5, 8, 11, 14, . . . , 3n – 1, . . .
the difference between successive terms is always 3.
Such a sequence is an arithmetic sequence or an arithmetic progression.
These sequences have the following property: The difference between successive terms is the same constant.
This constant is called the common difference. For the preceding sequence, the common difference is 3.
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Arithmetic Sequences
In general, an arithmetic sequence can be defined as follows.
Definition of an Arithmetic Sequence
Let d be a real number. A sequence an is an arithmetic sequence if
ai + 1 – ai = d for all positive integers i
The number d is the common difference for the sequence.
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Arithmetic Sequences
Example
3, 8, 13, 18, ... , 5n – 2, ...
11, 7, 3, –1, ... , –4n + 15, ...
1, 2, 3, 4, ... , n, ...
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Arithmetic Sequences
Consider an arithmetic sequence in which the first term is a1 and the common difference is d.
By adding the common difference to each successive term of the arithmetic sequence, we can find a formula for the nth term.
a1 = a1
a2 = a1 + d
a3 = a2 + d = a1 + d + d = a1 + 2d
a4 = a3 + d = a1 + 2d + d = a1 + 3d
Note the relationship between the term number and the coefficient of d. The coefficient is 1 less than the term number.
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Arithmetic Sequences
Formula for the nth Term of an Arithmetic Sequence
The nth term of an arithmetic sequence with common difference d is given by
an = a1 + (n – 1)d
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Example 2 – Find the nth Term of an Arithmetic Sequence
a. Find the 25th term of the arithmetic sequence whose first three terms are –12, –6, 0.
b. The 15th term of an arithmetic sequence is –3 and the first term is 25. Find the 10th term.
Solution:
a. Find the common difference:
d = a2 – a1
Use the formula
an = a1 + (n – 1)d with n = 25.
= –6 – (–12)
= 6
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Example 2 – Solution
a25 = –12 + (25 – 1)(6)
= –12 + 24(6)
= –12 + 144
= 132
b. Solve the equation an = a1 + (n – 1)d for d, given that
n = 15, a1 = 25, and a15 = –3.
–3 = 25 + (14)d
d = –2
cont’d
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Example 2 – Solution
Now find the 10th term.
an = a1 + (n – 1)d
a10 = 25 + (9)(–2)
= 7
n = 10, a1 = 25, d = –2
cont’d
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Arithmetic Series
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Arithmetic Series
Consider the arithmetic sequence given by
1, 3, 5, ... , 2n – 1, ...
Adding successive terms of this sequence, we generate a sequence of partial sums.
The sum of the first n terms of an arithmetic sequence is called an arithmetic series.
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Arithmetic Series
S1 = 1
S2 = 1 + 3 = 4
S3 = 1 + 3 + 5 = 9
S4 = 1 + 3 + 5 + 7 = 16
S5 = 1 + 3 + 5 + 7 + 9 = 25
Sn = 1 + 3 + ... + (2n – 1) = n2
The first five terms of this sequence are 1, 4, 9, 16, 25. It appears from this example that the sum of the first n odd integers is n2.
...
...
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Arithmetic Series
Shortly, we will be able to prove this result by using the following formula.
Formula for the nth Partial Sum of an Arithmetic Sequence
The nth partial sum of an arithmetic sequence an is
There is an alternative formula for the sum of n terms of an arithmetic sequence.
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Arithmetic Series
Alternative Formula for the nth Partial Sum of an Arithmetic Sequence
The nth partial sum of an arithmetic sequence an with common difference d is
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Example 3 – Find a Partial Sum of an Arithmetic Sequence
a. Find the sum of the first 100 terms of the arithmetic sequence given by ai = 2i – 1.
b. Find the sum of the first 50 terms of the arithmetic
sequence whose first three terms are 2,
Solution:
We could use either one of the formulas for the nth partial sum of an arithmetic sequence:
or
For a, we will use the first formula; for b, we will use the second formula.
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Example 3 – Solution
a. To use the formula we must first find
a1 and a100, given ai = 2i – 1.
ai = 2i – 1 ai = 2i – 1
a1 = 2(1) – 1 a100 = 2(100) – 1
= 1 = 199
= 50(200)
= 10,000
i = 1
i = 100
n = 100, a1 = 1, an = 199
cont’d
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Example 3 – Solution
b. To use the formula find d, the common difference.
cont’d
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Example 3 – Solution
cont’d
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Arithmetic Series
The first n positive integers 1, 2, 3, 4, ..., n are part of an arithmetic sequence with a common difference of 1, a1 = 1,
and an = n.
A formula for the sum of the first n positive integers can be found by using the formula for the nth partial sum of an arithmetic sequence.
Replacing a1 with 1 and an with n yields
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Arithmetic Series
This proves the following theorem.
Sum of the First n Positive Integers
The sum of the first n positive integers is given by
To find the sum of the first 85 positive integers, use n = 85.
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Geometric Sequences and Series
SECTION 11.3
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Geometric Sequences
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Geometric Sequences
Arithmetic sequences are characterized by a common difference between successive terms.
A geometric sequence is characterized by a common ratio between successive terms.
The sequence
3, 6, 12, 24, . . . , 3(2n –1), . . .
is a geometric sequence. Note that the ratio of any two successive terms is 2.
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Geometric Sequences
Definition of a Geometric Sequence
Let r be a nonzero constant real number. A sequence is a geometric sequence if
for all positive integers i.
The number r is called the common ratio.
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Geometric Sequences
Consider a geometric sequence in which the first term is a1 and the common ratio is r. By multiplying each successive term of the geometric sequence by the common ratio, we can derive a formula for the nth term.
a1 = a1
a2 = a1r
a3 = a2r = (a1r )r = a1r 2
a4 = a3r = (a1r 2)r = a1r 3
Note the relationship between the number of the term and the number that is the exponent on r. The exponent on r is 1 less than the number of the term.
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Geometric Sequences
With this observation, we can write a formula for the nth term of a geometric sequence.
The nth Term of a Geometric Sequence
The nth term of a geometric sequence with first term a1 and common ratio r is
an = a1r n –1
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Example 2 – Find the nth Term of a Geometric Sequence
Find the nth term of the geometric sequence whose first three terms are as follows.
a. b. 5,–10, 20, …
Solution:
a.
Thus
and a1 = 4.
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Example 1 – Solution
b.
Thus an = 5(–2)n –1.
cont’d
and a1 = 5.
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Finite Geometric Series
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Finite Geometric Series
The sum of the first n terms of a geometric sequence is a
finite geometric series.
Adding the terms of a geometric sequence, we can define the nth partial sum of a geometric sequence in a manner similar to that of an arithmetic sequence.
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Finite Geometric Series
Consider sequence of partial sums for the geometric sequence 1, 2, 4, 8, . . . , 2n – 1, . . .
S1 = 1
S2 = 1 + 2 = 3
S3 = 1 + 2 + 4 = 7
S4 = 1 + 2 + 4 + 8 = 15
Sn = 1 + 2 + 4 + 8 + . . . + 2n – 1
The first four terms of the sequence of partial sums are 1, 3, 7, and 15.
. . .
. . .
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Finite Geometric Series
To find a general formula for Sn, the nth term of the sequence of partial sums of a geometric sequence, let
Sn = a1 + a1r + a1r 2 + . . . + a1r n – 1
Multiply each side of this equation by r.
Sn = a1 + a1r + a1r 2 + . . . + a1r n – 2 + a1r n – 1
rSn = a1r + a1r 2 + . . . + a1r n – 2 + a1r n – 1 + a1r n
Subtract the two equations.
Sn – rSn = a1 – a1r n
Sn(1 – r) = a1(1 – r n)
Factor out the common factors.
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Finite Geometric Series
This proves the following theorem.
Formula for the nth Partial Sum of a Geometric Sequence
The nth partial sum of a geometric sequence with first term a1 and common ratio r is
r 1
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Example 3 – Find a Partial Sum of a Geometric Sequence
a. Find the sum of the first four terms of the geometric sequence 5, 15, 45, . . . , 5(3)n – 1, . . . .
b. Evaluate the finite geometric series .
Solution:
a. We have a1 = 5, r = 3 and n = 4.
Thus
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Example 3 – Solution
b. When n = 1, a1 = 3. The first term is 3. The second term
is . Therefore, the common ratio is . Thus
cont’d
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Infinite Geometric Series
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Infinite Geometric Series
Following are two examples of geometric sequences for which | r | < 1.
Note that when the absolute value of the common ratio of a geometric sequence is less than 1, the terms of the geometric sequence approach zero as n increases.
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Infinite Geometric Series
We write, for | r | < 1, | r |n 0 as n .
Consider again the geometric sequence
The nth partial sums for n = 3, 6, 9, and 12 are given in Table 11.1, along with the values of r n.
Table 11.1
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Infinite Geometric Series
As n increases, Sn is closer to 4 and r n is closer to zero. By finding more values of Sn for larger values of n, we would find that Sn 4 as n .
As n becomes larger Sn is the nth partial sum of ever more terms of the sequence. The sum of all the terms of a sequence is called an infinite series.
If the sequence is a geometric sequence, we have an infinite geometric series.
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Infinite Geometric Series
Sum of an Infinite Geometric Series
If an is a geometric sequence with | r | < 1 and first term a1, then the sum of the infinite geometric series is
An infinite series is represented by
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Example 4 – Find the Sum of an Infinite Geometric Series
Evaluate the infinite geometric series .
Solution:
The general term is . To find the first term, let
n = 1.
Then
The common ratio is
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Example 4 – Solution
cont’d
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Consider the repeating decimal
The right–hand side is a geometric series with
and common ratio Thus
Infinite Geometric Series
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Infinite Geometric Series
The repeating decimal
We can write any repeating decimal as a ratio of two integers by using the formula for the sum of an infinite geometric series.
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