Math 2224 Multivariable Calculus – Chapter 9 Infinite Sequences and Series
Sec. 9.1: Sequences
I.
KEY
Sequence
A. Definitions
1. Defn: A sequence is a list of numbers written in a definite order
where
a1, a2 , a3 ,..., an ,...
a1 = 1 term, a2 = 2nd term, a3 = 3rd term, an = nth term
st
+
2. An infinite sequence of numbers is a function whose domain is
You can denote the sequence by its nth term
B. Examples
1.
{an } or {an }n=1 .
(the set of positive integers).
an = n
¥
ì (-1)n+1 ü
í
ý
î n þn=1
1 1 1
2. 1,- , ,- ,...
2 3 4
an .
¥
{n}¥n=1
1,2, 3,..., n,...
and n=index of
3.
an = cos(p n)
4.
an =
3n + 1
n+2
4 7 10 13
, ,
,
,...
3 4 5 6
5.
an =
en
2n
e e2 e 3 e 4
,
,
,
,...
2 4 8 16
-1,1,-1,1,...
(-1)n+1
an =
n
{(-1) }
n
C. Graphical Representation of Sequences
1. There are two ways to represent sequences graphically. The first method represents the first
few terms on the real axis. The second method represents the first few terms as points on the
graph of the function defining the sequence. The function is defined only on integer inputs and
the points on the graph are (1,a1 ), (2,a2 ), (3,a3 ),..., (n,an ),...
2. Example - # 2 from above
a2
II.
a4
a5 a3
Convergence and Divergence
A. Definition
1. Defn: The sequence
a1
{an } converges to the number L if for every positive number e
corresponds an integer N such that "n ,
n > N Þ an - L < e .
there
If no such number L exists, we say that diverges.
2. A sequence
If
lim an = L , L is the limit of the sequence.
{an } converges to L, if n®¥
{an } does not converge, then it diverges.
3. The sequence
{an } diverges to infinity if for every number M there is an integer N such that
"n > N , an > M .
If this condition holds we write
lim an ® ¥ or an ® ¥ .
n®¥
"n > N , an < m , then we say
lim an ® -¥ or an ® -¥ .
Similarly if for every number m there is an integer N such that
{an } diverges to negative infinity and write n®¥
B. Limit Theorems
1. Thm 1: Let
{an } and {bn } be sequences of real numbers and let A and B be real numbers.
The following rules hold if
1. Sum/Difference Rule:
lim an = A and lim bn = B
n®¥
lim(an ± bn ) = A ± B
n®¥
2. Constant Multiple Rule:
3. Product Rule:
n®¥
lim(kan ) = kA (any number k)
n®¥
lim(an ×bn ) = A × B
n®¥
an A
=
n®¥ b
B
n
if B ¹ 0
4. Quotient Rule:
lim
5. Constant Rule:
lim(k) = k (any number k)
n®¥
2. Thm 2 – Squeeze Thm for Sequences:
Let
{an }, {bn } and {cn } be sequences of real numbers. If an £ bn £ cn holds "n beyond
some index N, and if
lim an = lim cn = L, then lim bn = L also.
n®¥
n®¥
n®¥
3. Thm 3 – The Continuous Function Thm for Sequences
Let
{an } be a sequence of real numbers. If an ® L and if f is a function that is continuous at
L and defined at all an , then f (an ) ® f (L).
4. Thm 4: Suppose that
numbers such that
f(x) is a function defined "x ³ n0 and that {an } is a sequence of real
an = f (n) for n ³ n0 .
lim f (x) = L Þ lim an = L .
Then
x®¥
n®¥
Thm 4 allows us to use L’Hopital’s Rule to find limits of some sequences.
D. Examples: Do the following sequences converge?
1.
1,2, 3,..., n,...
2.
1 1 1
1,- , , - ,...
2 3 4
lim n ® ¥; DNE Þ {n} diverges
n®¥
-1 (-1)n-1 1
-1
1
(-1)n-1
<
< and lim = lim = 0, by the Squeeze Th m for Sequences lim
=0 ;
n®¥ n
n®¥ n
n
n
n
n
n®¥
ì (-1)n-1 ü
í
ý converges to 0
î n þ
3.
an = cos(p n)
ì1 if n is even
lim cos(p n) DNE b/c cos(p n) = í
n®¥
î-1 if n is odd
4.
an =
3n + 1
n+2
Þ {cos(p n)} diverges
3n + 1
3 + 1n
= lim
=3 ; sequence converges to 3
n®¥ n + 2
n®¥ 1 + 2
n
lim
or use L’Hopital’s Rule
en
5. an = n
2
n
en
æ eö
lim n =lim ç ÷ > 1n (always increasing)
n®¥ 2
n®¥ è 2 ø
ì en ü
Þ í n ý diverges
î2 þ
6.
an =
sin(2n)
1+ n
-1 £ sin(2n) £ 1
-1
sin(2n)
1
£
£
1+ n 1+ n 1+ n
-1
sin(2n)
1
£ lim
£ lim
n®¥ 1 +
n n®¥ 1 + n n® ¥ 1 + n
lim
sin(2n)
sin(2n)
£ 0 Þ lim
= 0 by Squeeze Thm
n®¥ 1 +
n®¥ 1 +
n
n
Sequence converges to 0
0 £ lim
E. Commonly Occurring Limits
The following six sequences converge to the limits listed below:
1.
lim
n®¥
ln n
=0
n
1
3.
lim x n = 1
2.
(x > 0)
n®¥
4.
n
xö
æ
x
5. lim ç 1 + ÷ = e
n®¥ è
nø
(any x)
In formulas (3) through (6), x remains fixed as
III.
Bounded Sequences
A. Definitions
lim n n = 1
n®¥
lim x n = 0
( x < 1)
xn
=0
n®¥ n!
(any x)
n®¥
6. lim
n ® ¥.
1. A sequence
{an } with the property an £ an+1
"n is called a nondecreasing sequence.
2. A sequence
{an } with the property an < an+1
"n ³ 1 is called a monotone increasing
{an } with the property an > an+1
"n ³ 1 is called a monotone decreasing
{an } with the property an ³ an+1
"n is called a nonincreasing sequence.
sequence.
3. A sequence
sequence.
4. A sequence
{an } is bounded if there exists numbers M and N such that N £ an £ M "n .
A sequence { an } is bounded from above if there exists a number M such that an £ M "n .
The number M is an upper bound for { an }. If M is an upper bound for { an } but no number
less that M is an upper bound for { an }, then M is the least upper bound for { an }.
5. A sequence
6.
{an } is bounded from below if there exists a number N such that N £ an "n .
The number N is a lower bound for { an }. If N is a lower bound for { an } but no number
greater that N is a lower bound for { an }, then L is the greatest lower bound for { an }.
7. A sequence
B. Theorem 6 – The Nondecreasing Sequence Thm
A nondecreasing sequence of real numbers converges iff it is bounded from above. If a
nondecreasing sequence converges, it converges to its least upper bound.
Thm 6 implies that a nondecreasing sequence converges when it is bounded from above. It
diverges to infinity if it is not bounded from above.
C. Example
Determine the least upper bound and the greatest lower bound, if they exist, if
4 5 6 7 8 9
2,1, , , , , , ,...
5 7 9 11 13 15
Least Upper Bound=a1=2
Greatest Lower Bound=
IV.
1
n +1 1
, b/c lim
=
n®¥ 2n - 1
2
2
Extra Examples – Determine if the following sequences converge
1.
an = (-1)n sin( 1n )
lim(-1)n sin( 1n ) = 0 Þ sequence converges to 0
n®¥
2.
3.
an =
n
1+ n
lim
n®¥
n L'H
1
= n®¥
lim
-1 = lim 2 n ® ¥ Þ sequence diverges
1+ n
- 12 n 2 n®¥
an = (1 + an )n
an =
n +1
2n - 1
ln(1+ an )n
lim (1+ an )n = n®¥
lim e
n®¥
æ
L'H
= n®¥
lim e
4.
an =
0£
nln(1+ an )
= n®¥
lim e
= n®¥
lim e
ln(1+ an )
1
n
ö
1 ç -a ÷
(1+ an ) ççè n2 ÷÷ø
-1
n2
= n®¥
lim e
a
(1+ an )
= ea Þ sequence converges to ea
3 + (-1)n
n2
3 + (-1)n
4
£ 2
2
n
n
3 + (-1)n
4
£ lim 2
2
n®¥
n® ¥ n
n
lim 0 £ lim
n®¥
3 + (-1)n
3 + (-1)n
0 £ lim
£ 0 Þ lim
= 0 by the Squeeze Thm
n®¥
n®¥
n2
n2
Þ sequence converges to 0