Math 1432
Bekki George
[email protected]
639 PGH
James West
[email protected]
Office Hours:
(by appointment and check on CASA)
Class webpage:
www.casa.uh.edu
n
æ 2ö
Consider the sequence defined by an = ç ÷ . (n starts at 1)
è 3ø
a. Write the first five terms of the sequence.
b. Determine the limit of the sequence.
c. Let bn =
an+1
. Write the first five terms of this sequence.
an
d. Determine the limit of bn .
n
æ -3 ö
Consider the sequence defined by an = ç ÷ . (n starts at 1)
è 2ø
a. Write the first five terms of the sequence.
b. Determine the limit of the sequence.
c. Let bn =
an+1
. Write the first five terms of this sequence.
an
d. Determine the limit of bn .
Are the following increasing, decreasing, or not monotonic?
3n + 4
an =
2n + 5
3 + ( -1)
an =
n
n +1
an =
5n + 3
n
Give an upper bound for the set of negative real numbers.
Give a lower bound for the set of negative real numbers.
Give the LUB and GLB for the set of negative real numbers.
ì ( -1 )
Give the LUB and GLB of ïí
n
ïî
¥
ü
ï
ý
ïþ
n=3
n
¥
ì
ü
Determine whether ïí ln æ 2n - 1 ö ïý is bounded.
çè 3n + 7 ÷ø
ïî
ïþ n=1
Sequences can be defined recursively: one or more terms are
given explicitly; the remaining ones are then defined in terms of
their predecessors. Give the first six terms
of the sequence and then give the nth term.
a1 = 1; an+1 = ½ an + 1.
Popper 20
n 
  1
1. Give the limit (if it exists) of 
 n
a. –1
b. 0
c. 1
d. 1/2

 .
n1

1  sin  n  
2. Give the limit (if it exists) of 

n

n1
a. –1
b. 0
c. 1
d. 1/2
3. Give the LUB for {x | x2 – 2x < 3}
a.
b.
c.
d.
e.
3
–3
1
–1
none of these
With sequences, we are concerned with limits of sequences as n approaches
infinity.
A sequence that has a limit is said to be convergent.
A sequence that has no limit is said to be divergent.
Every convergent sequence is bounded and every unbounded sequence is
divergent.
“The sequence converges”
means
“The sequence has a limit”.
“The sequence diverges”
means
“The sequence does not have a limit”.
Important Limits:
1
®0
as n ® ¥
xn
B. For each real x,
®0
n!
as n ® ¥
A. For each a > 0,
n
a
C. If |x| < 1, then x n ® 0
as n ® ¥
ln n
D.
® 0 as n ® ¥
n
E. If x > 0, then x
1
n
®1
as n ® ¥
F. n
1
n
®1
as n ® ¥
n
x ö
æ
G. For each real x, ç 1 + ÷ ® e x as n ® ¥
è
n ø
Examples:
Give the limit of
{ ( -1 ) }
n
¥
n=1
¥
Give the limit of
ì 2n - 6 ü
í
ý
2
î 3n + 2 þ n=1
Give the limit (if it exists) of
{ ln ( n + 1 ) - ln ( n ) } n=1
Give the limit (if it exists) of
{ ln ( 2n + 1 ) - ln ( n ) } n=1
¥
¥
Give the limit (if it exists) of
ìï ln ( n + 1 )
í
n
ïî
¥
üï
ý
ïþ n=1
Popper20
 l n  n  2 
4. Give the limit of the sequence: 
2 n

a.
b.
c.
d.
e.
DNE
1
0
3
e



 n1
Give the limit (if it exists) of
ìï 3
í n
ïî 4
Give the limit (if it exists) of
ìï
ín
ïî
n
1
n+2
¥
üï
ý
ïþ n=1
¥
üï
ý
ïþ n=1
ìï
Note: í e n
ïî
( )
1
n
¥
¥
üï
®e
ý
ïþ n=1
ìï
üï
í ( stuff ) ý
ïî
ïþ n=1
may not go to 1 as n approaches infinity if “stuff” overpowers the exponent.
Be careful!
1
n
Give the limit (if it exists) of
¥
ìï æ
1 ö üï
í ç 1- ÷ ý
n ø ï
ïî è
þ n=1
n
¥
ìï æ
2 ö üï
5. Give the limit of í ç 1 - ÷ ý
n ø ï
ïî è
þ n=1
n
a.
b.
c.
d.
e.
1
e2
e
e-2
none of these
ìï 3n 3 - 2 n + 1
6. Give the limit of í
2
3
ïî 1000 n - n + 3
a. –3
b. 3
c. 3/1000
d. DNE
7. Give the limit of
a.
b.
c.
d.
–1
1
0
DNE
{ cos ( np ) }
¥
n=1
¥
üï
ý
ïþ n=1
8. Give the limit (if it exists) of the sequence
a.
b.
c.
d.
e.
3
1
0
9
DNE
¥
ì
ü
í3 ý
î
þ1
2
n