8.1 Sequences
Dec 27, 2015
Quick Review
Let ( ) . Find the values of .4
1. (5)
2. (-1)
Evaluate the expression 1 for the given values of
, , and .
3. -2, 2, 3
4. 1, 2, 2
xf x f
xf
f
a n d
a n d
a n d
a n d
9
5
3
1
15
Quick Review
-1
2
2
0
lim
lim
Evaluate the expression for the given values of , , .
15. , 2, 3
26. 2, 1.5, 4
Find the value of the limit.
2 27.
4 1sin 4
8.
n
x
x
ar a r and n
a r n
a r n
x
x xx
x
2
75.6
2
1
4
What you’ll learn about Defining a Sequence Arithmetic and Geometric Sequences Graphing a Sequence Limit of a Sequence
Essential QuestionHow can we use calculus to define andevaluate sequences?
Defining a Sequence
1 2 3 1
1 2 3
A is a list of numbers written in an explicit order.
For example: , , ,..., ,... , where is the first
and is the of the sequence.
Let , , ,..., ,... be a funct
n
n n
n
n
a
a a a a a a
a
a a a a
sequence
term
nth term
1 2 3
ion with domain the set of positive
integers and range , , ,..., ,... . If the domain is finite, then
the sequence is a . If the domain is infinite, then
the sequence is an
na a a a
finite sequence
infinite .sequence
Example Defining a Sequence Explicitly1. Find the first four terms and the 100th term of the sequence {a
n} where
.
2
12
n
an
n Set n equal to 1, 2, 3, 4, and 100.
21
12
1
1
a3
1
22
12
2
2
a6
1
23
12
3
3
a11
1
24
12
4
4
a18
1
2100
12
100
100
a002,10
1
Example Defining a Sequence Recursively
2. Find the first three terms and the 7th term of the sequence defined recursively by the conditions: b
1 = 4 and b
n = b
n – 1 – 2 for all n > 2.
41 b
2122 bb 21 b 24 2
2133 bb 22 b 22 0
2177 bb 26 b 26 8
Arithmetic SequenceA sequence {a} is an arithmetic sequence if it can be written in the form {a, a + d, a + 2d, . . . , a + (n – 1)d, . . .} for some constant d. The number d is the common difference.
.2 allfor 1 ndaa nn
Each term in an arithmetic sequence can be obtained recursively from its preceding term by adding d:
Example Defining Arithmetic Sequences3. Given the arithmetic sequence: – 3, 1, 5, 9, . . . find
a. the common difference,
b. the ninth term,
c. a recursive rule for the nth term,
d. an explicit rule for the nth term.
. is differencecommon The a. 12 aa 31 4 dnaan 1 b. 1
9a 3 19 4 29:is rule recursive The c. ,31 a 41 nn aa
dnaan 1 :is ruleexplicit The d. 1
na 3 1 n 4 74 n
Geometric SequenceA sequence {a} is an geometric sequence if it can be written in the
form {a, a . r, a . r2, . . . , a . r n – 1 , . . .} for some nonzero constant r. The number r is the common ratio.
.2 allfor 1 nraa nn
Each term in an geometric sequence can be obtained recursively from its preceding term by multiplying by r:
Example Defining Geometric Sequences4. Given the geometric sequence: 1, – 3, 9, – 27, . . . find
a. the common ratio,
b. the tenth term,
c. a recursive rule for the nth term,
d. an explicit rule for the nth term.
. is ratiocommon The a.1
2
a
a
1
3 31
1 b. nn raa
9a 1 3 110 683,19:is rule recursive The c. ,11 a 13 nn aa
11 :is ruleexplicit The d. n
n raa
na 1 3 1n 13 n
Example Constructing a Sequence5. The second and fifth term of a geometric sequence are – 6 and 48,
respectively. Find the first term, common ratio and an explicit rule for the nth term.
121
151
ra
ra1
1
41
ra
ra
6
48
3r 82r
61 ra
62 1 a
31 a1
1 :is ruleexplicit The nn raa
na 3 2 1n 11 231 nn
Example Graphing a Sequence Using Parametric Mode
6. Draw a graph of the sequence {an} with . . . 3, 2, ,1 ,1
1
nn
na n
n
Change the mode on your calculator to parametric and dot.
T
TT 11Y T,XLet 1T1T
1T ,20T ,1T stepmaxmin Set your window for the following:
2X ,20X ,0X sclmaxmin 1Y ,2Y ,2Y sclmaxmin
Example Graphing a Sequence Using Sequence Graphing Mode
7. Graph the sequence defined recursively by b1 = 4 and b
n = b
n – 1 + 2 for
all n > 2.Change the mode on your calculator to sequence and dot.Replace b
n by u(n).
Select nMin = 1, u(n) =u(n – 1) + 2, and u(nMin) = {4}.
Example Graphing a Sequence Using Sequence Graphing Mode
7. Graph the sequence defined recursively by b1 = 4 and b
n = b
n – 1 + 2 for
all n > 2.
Set nMin = 1, uMax = 10, PlotStart = 1, PlotStep = 1, and graph in the [0, 10] by [– 5, 25] viewing window.
Limit Let L be a real number. The sequence a has limit L as n approaches ∞
if, given any positive number , there is a positive number M such that for all n > M we have . Lan
We write Lann
lim and say that the sequence converges to L.
Sequences that do not have limits diverge.
Properties of Limits If L and M are real numbers and
1. Sum Rule:
Lann
lim and , lim Mbn
n
then
MLba nnn
lim
2. Difference Rule: MLba nnn
lim
3. Product Rule: MLba nnn
lim
4. Constant Multiple Rule: Lccann
lim
5. Quotient Rule: 0,lim
MM
L
b
a
n
n
n
Example Finding the Limit of a Sequence8. Determine whether the sequence converges or diverges. If it
converges, find its limit.
n
nan
12
Graph it, changing the mode to parametric and dot.
Find the limit analytically, using the Properties of Limits:
n
nn
12lim
nn
12lim
nnn
1lim2lim
2 0 2
The Sandwich Theorem for Sequencesand if there is an integer N for whichLca n
nn
n
lim lim If
LbNncba nn
nnn
lim then , allfor
Absolute Value Theorem .0 lim then ,0 lim If . sequence heConsider t
nn
nn
n aaa
Example Using the Sandwich Theorem to find the Limit of a Sequence
9. Show that the following sequence converges and find its limit.
n
nan
cos 1cos x
n
ncos n
ncos
n
1
n
1
n
ncos
n
1
nn
1lim 0
nn
1lim 0
n
nn
coslim 0