Section 8.1: Sequences
A sequence is a function whose domain is ℕ.
A sequence is a list.
sn 
or sometimes
{sn }
s
s
( 1/n ) converges to 0
Idea:
When n gets big, 1/n gets small
nn
n!
3n
en
n4
n2
n
√n
∜n
ln n
 3n2  4 
 2

 n  5n 
Converges to 3
 3n3  4 
 2

 n  5n 
Diverges to infinity
 3n 2  4 
 6

 n  5n 
Converges to 0
 3 ln n  4 


 n5 n 
Converges to 0
 5e n  n 
 2

n 2 n 
Diverges to ∞
Suppose (sn ) converges to s and
(tn ) converges to t.
lim n ( sn  tn )  s  t
lim n ksn  ks
lim n sntn  st
lim n
sn s
 ,
tn t
if tn  0, t  0
Suppose an  bn  cn and
bn  L
Then lim
n 
lim an  lim cn  L
n 
n 
 1
lim 1  
k 
 k
k
Notice that we have k’s raised to the kth power.
This is a job for L’Hopital’s Rule!
 1
lim 1  
k 
 k
k
First, we need to commit algebra.
We change the problem by taking the log.
 1
ln 1  
 k
k
 1
k ln 1  
 k
 1
ln 1  
 k
1
k
 1
ln 1  
x

lim
x 
1
x
 1
ln 1  
k

lim
k 
1
k
1
 1  ( x 2 )
1  
x

lim
2
x 
x
lim
x 
1
 1
1  
 x
1
k
 1
lim ln 1    1
k 
 k
 1
lim 1  
k 
 k
Remember we changed the problem
by taking the log!
ln ( what we want)  1
what we want  e
1
k
 1
lim 1    e
k 
 k
k
lim k
1/ k
k 
Again, we need to commit algebra.
We change the problem by taking the log.
ln k
1
ln k
k
1/ k
ln k
k
ln k
lim
 0 by the tower of power!
k 
k
lim
x 
1
x
1
 0
ln x
lim
x 
x
lim ln k
k 
1/ k
 0
lim k 1/ k
k 
Remember we changed the problem
by taking the log!
ln ( what we want)  0
what we want  e0  1
lim k
k 
1/ k
1
k
 1
lim 1    e
k 
 k
lim k
k 
lim
k 
k
1/ k
1
k 1
an is increasing if an 1  an for all n
an is strictly increasing if an 1  an for all n
an is decreasing if
an 1  an for all n
an is strictly decreasing if an 1  an for all n
an is monotone if it is either increasing or decreasing
an is bounded above by M if an  M for all n
an is bounded below by M if an  M for all n
an
is bounded if it is bounded both above and below
An increasing sequence converges if and only if it is
bounded above.
A decreasing sequence converges if and only if it is
bounded below.
A monotone sequence converges if and only if it is
bounded.