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Sequences and Series
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12.1
Sequences and Summation
Notation
Copyright © Cengage Learning. All rights reserved.
Objectives
► Sequences
► Recursively Defined Sequences
► The Partial Sums of a Sequence
► Sigma Notation
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Sequences and Summation Notation
Roughly speaking, a sequence is an infinite list of numbers.
The numbers in the sequence are often written as
a1, a2, a3, . . . . The dots mean that the list continues
forever. A simple example is the sequence
We can describe the pattern of the sequence displayed
above by the following formula:
an = 5n
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Sequences
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Sequences
Any ordered list of numbers can be viewed as a function
whose input values are 1, 2, 3, . . . and whose output
values are the numbers in the list. So we define a
sequence as follows:
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Example 1 – Finding the Terms of a Sequence
Find the first five terms and the 100th term of the sequence
defined by each formula.
(a) an = 2n – 1
(b) cn = n2 – 1
(c)
(d)
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Example 1 – Solution
To find the first five terms, we substitute n = 1, 2, 3, 4, and
5 in the formula for the nth term.
To find the 100th term, we substitute n = 100. This gives
the following.
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Example 2 – Finding the nth Term of a Sequence
Find the nth term of a sequence whose first several terms
are given.
(a)
(b) –2, 4, –8, 16, –32, . . .
Solution:
(a) We notice that the numerators of these fractions are the
odd numbers and the denominators are the even
numbers. Even numbers are of the form 2n, and odd
numbers are of the form 2n – 1 (an odd number differs
from an even number by 1).
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Example 2 – Solution
cont’d
So a sequence that has these numbers for its first four
terms is given by
(b) These numbers are powers of 2, and they alternate in
sign, so a sequence that agrees with these terms is
given by
an = (–1)n2n
You should check that these formulas do indeed
generate the given terms.
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Recursively Defined Sequences
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Recursively Defined Sequences
Some sequences do not have simple defining formulas like
those of the preceding example.
The nth term of a sequence may depend on some or all of
the terms preceding it.
A sequence defined in this way is called recursive. Here is
an example.
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Example 4 – The Fibonacci Sequence
Find the first 11 terms of the sequence defined recursively
by F1 = 1, F2 = 1 and Fn = Fn –1 + Fn –2
Solution:
To find Fn, we need to find the two preceding terms, Fn –1
and Fn –2. Since we are given F1 and F2, we proceed as
follows.
F3 = F2 + F1 = 1 + 1 = 2
F4 = F3 + F2 = 2 + 1 = 3
F5 = F4 + F3 = 3 + 2 = 5
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Example 4 – Solution
cont’d
It’s clear what is happening here. Each term is simply the
sum of the two terms that precede it, so we can easily write
down as many terms as we please.
Here are the first 11 terms:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, . . .
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Recursively Defined Sequences
The sequence in Example 4 is called the Fibonacci
sequence, named after the 13th century Italian
mathematician who used it to solve a problem about the
breeding of rabbits.
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The Partial Sums of a Sequence
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The Partial Sums of a Sequence
In calculus we are often interested in adding the terms of a
sequence. This leads to the following definition.
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Example 5 – Finding the Partial Sums of a Sequence
Find the first four partial sums and the nth partial sum of
the sequence given by an = 1/2n.
Solution:
The terms of the sequence are
The first four partial sums are
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Example 5 – Solution
cont’d
Notice that in the value of each partial sum, the
denominator is a power of 2 and the numerator is one less
than the denominator.
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Example 5 – Solution
cont’d
In general, the nth partial sum is
The first five terms of an and Sn are graphed in Figure 7.
Graph of the sequence an and the sequence of partial sums Sn
Figure 7
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Sigma Notation
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Sigma Notation
Given a sequence
a1, a2, a3, a4, . . .
we can write the sum of the first n terms using summation
notation, or sigma notation. This notation derives its
name from the Greek letter  (capital sigma, corresponding
to our S for “sum”). Sigma notation is used as follows:
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Sigma Notation
The left side of this expression is read, “The sum of ak from
k = 1 to k = n.”
The letter k is called the index of summation, or the
summation variable, and the idea is to replace k in the
expression after the sigma by the integers 1, 2, 3, . . . , n,
and add the resulting expressions, arriving at the right side
of the equation.
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Example 7 – Sigma Notation
Find each sum.
Solution:
= 12 + 22 + 32 + 42 + 52
= 55
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Example 7 – Solution
cont’d
= 5 + 6 + 7 + 8 + 9 + 10
= 45
=2+2+2+2+2+2
= 12
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Example 7 – Solution
cont’d
We can use a graphing calculator to evaluate sums. For
instance, Figure 8 shows how the TI-83 can be used to
evaluate the sums in parts (a) and (b) of Example 7.
Figure 8
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Sigma Notation
The following properties of sums are natural consequences
of properties of the real numbers.
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