Non- Convergent Sequences.
A sequence that converges to L gets as close to
L as we please. In our language it means that if
we take any neighborhood of L, all the terms of
the sequence will be in that neighborhood but a
finite number of them. Sequences whose values
do not approach to certain value are called nonconvergent. These are several types of non-convergent sequences. First let’s concentrate
on the ones we call divergent sequences. These are two examples.
1) The sequence nn1 gets as positive as we please. It is, their distance from zero (or

from any number L one chooses) gets as large as we want for almost all the terms of
the sequence. We can see it first from its graph. In this example we can observe that
any A-neighborhood of zero, no matter how big it is, will have an infinite number of
positive terms outside it. From the graph we can see that for A  40 , d ( n,0)  40 is
satisfied only for n  40 . In other words, starting at 40 there is an infinite number of
terms of the sequence, all positive, outside the neighborhood. The same reasoning can
be done for any A-neighborhood one chooses.
How many terms of the sequence are there inside a 300-neighborhood of zero?
A formal argument will guarantee that for any value A  0 , there is N, such
that an  A, for n  N . In other words, only a finite number of terms are in any given
neighborhood of 0, no matter how big we choose the neighborhood. Or equivalently
given an A-neighborhood of zero, starting at some N all the terms of the sequence are
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outside it. Sequences that behave this way are said to diverge to  . In our example we
have that Limn  n   .
EXERCISE:
Show that the following sequences diverge to  . Choose the A-neighborhood as big as
you want.
i)
an  n , n  1
ii) bn  Ln(n  3), n  3
iii) c n 
n2 1
, n  2 . Algebraically we can show this sequence diverges too.
n 1
Use the technique from the previous section.
2) The sequence a n   n 2 n 1 gets as negative as we please. It means that even though

the terms are negative they are as far away from zero as we please. Let’s look at the

graph below.
For any A-neighborhood of zero, there will be an infinite number of terms of the
sequence outside it. For instance, when A  2000 , how many terms are there inside the
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2000-neighborhood of zero? This is the same as solving the inequality d (n 2 ,0)  2000 ,
or  n 2  2000 . Verify your answer graphically.
This sequence is said to diverge to  . It is denoted by Limn  n 2  
The next examples refer to sequences which neither converge nor diverge.
3) Consider the sequence
1 n 
n
2

n1
in the 2000-neighborhood of zero. From the
graph we can intuitively
say that it is neither
convergent nor divergent.
Why it does not converge?
Justify it.
It does not diverge, since any
neighborhood of zero, will
have an infinite number of terms, positive and negative values, outside it.
Find the values of n for which 1 n  2000
n
2
In the case of a 300,000neighborhood of zero, there is
an infinite number of terms of
the sequence outside it. Which
ones are outside the
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neighborhood? To find it, solve 1 n 2  300000
n
If we take A to be any positive number, for which values of n will the sequence
satisfy 1 n 2  A ?
n
This sequence does not converge, but does not diverge either since its values are going to
positive and negative infinity.
4) The sequence
1 2 
n
8

n1
does neither converge nor diverge. To convince yourself
about it, answer the following questions:
a) If the sequence converged, which number would be the candidate to be the limit?
b) Why it does not diverge?
Definition of Non-Convergence of a Sequence: A sequence {an} does not converge
to a limit L if there exists ε > 0 such that for each N there is an infinite number of
values n  N for which an  L  
Definition of Divergence of a Sequence: A sequence {an} diverges to ∞ (an → ∞) if
for every A  0 there exists N such that an  A , a n  A for any n  N . A sequence
{an} diverges to -∞ (an → - ∞) if for every A > 0 there exists N such that an  A ,
an   A
HOMEWORK
Answer each of the following questions using a couple of sentences. Illustrate your
answers with examples if you wish
1. What is the main difference between both of these definitions?
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2. Give an example, different than the ones given above, of a non-convergent
sequence that is non-divergent.
3. Could a non-convergent sequence be divergent? Why or why not?
4. Could a divergent sequence be non-convergent? Why or why not?
5. Could a convergent sequence be divergent or non-convergent? Why or why
not? Try to generate examples.
6. In the case of a convergent sequence, explain how reducing the size of ε
affects the corresponding N (i.e. if you first choose an ε and find an N, and
then choose a smaller ε than the first one, what happens to the N?
7. Produce arguments to show that the sequence
1 n does neither converge
n

1
nor diverge.
 
8. Show that Log n 2 , n  10 diverges.
9. Explain why the sequence
1 
n 
n1
neither diverges nor converges.
10.For each of the following sequences determine which one converges,
diverges, or neither. Justify your conclusion.
a.
 n

n10
 n n  1

b. 1

n n1

 1

c.  n1  100
3
 n 0
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PROPERTIES OF CONVERGENT SEQUENCES
Suppose that Lim a n  A, Lim bn  B
n 
n
1. Lim (a n  bn )  A  B
n 
2 . Lim (c * a n )  c * A , c a constant
n 
3. Lim
n
an A
 ,B  0
bn B
4. Lim a n * bn  AB . As a consequence Lim ( an ) k  Ak , k  
n 
n
Also notice that when Lim a n  , Lim bn   , Lim (a n  bn )   . It is the sum of two
n 
n
n 
divergent sequences diverges.
The following cases are indeterminate since result can be infinite, or a constant. One has
to analyze each case separately and try to eliminate the indeterminate.
Case I:    These are some examples which show why it is indeterminate.
Lim (n 2  n)  
n
Lim (n  n 2 )  
n
Lim [(1  n)  (5  n)]  4
n 
Case II: 0 *  ,
 0
,
 0
1
n

Lim (n * )Form  * 0  Lim  Form   1
n 
n  n
n


2
n 1

Lim
 form   
n 

n
1
0
2n

n
Lim  n  form   Lim
  Dominance of 2 n over n
n  2
0  n  n

e n  0 
e n1
Lim ( n1)    Lim n  e
n  e
 0  n e
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