

Sequences
and limits 
Infinite
sequences 
Sequences notation 
the
rule for the nth
term of a sequence 
Graphing the terms of a sequence on the number line 
The limit of a sequence 
The definition of the limit of a sequence 
Convergence of a sequence 





Infinite
Sequences 
An
infinite sequence is an ordered list of real numbers indexed by the
natural numbers n
Î
N
denoted {a_{n}} 
a_{1},
a_{2}, a_{3}, . . . , a_{n},
. . . 
where
by a_{n}
given is a rule to calculate the n^{th}
term of the sequence. 

Graphing the terms of a sequence
on the number line 
Thus,
for example; 
(1)
a_{n}
= n
for n =
1, 2, 3,. . .
gives the sequence,
1, 2, 3, 4, 5,.
. . 
shown
on the number line 


(2) 



for n =
1, 2, 3,. . .
gives the sequence, 




shown
on the number line 


(3) 



for n =
1, 2, 3,. . .
gives the sequence, 




shown
on the number line 


(4) 



for n =
1, 2, 3,. . .
gives the sequence, 




shown
on the number line 


(5) 



for n =
1, 2, 3, . . . gives the sequence, 




shown
on the number line 


Observe that, the first and the third sequence shown above both increase, the second and the fourth decrease,
and that the
terms of the fifth sequence oscillate (alternate) from the left to the right approaching closer and
closer to 1. 

The limit of a sequence 
The definition of the limit of a
sequence 
A
number L is
called the limit of a sequence {a_{n}}
if
for every positive number e there exist a natural number 
n_{0}
such that if
n
> n_{0},
then 
a_{n}

L

< e. 
That
is, a
number L is the limit of a sequence
if the distance between the term a_{n}
and L
becomes arbitrary small
by choosing n
large enough (see the examples above). The n_{0}
denotes the value of the index n
starting from which the distance a_{n}

L
becomes smaller than the given e. 
Since the value of n_{0}
depends on the size of e
it is usually written as n_{0}(e). 

Convergence of a sequence 
Therefore,
if a sequence {a_{n}}
has a limit L
we write 


or 
a_{n}
®
L
as n
®
oo



and
we say that that a sequence a_{n}
has the limit L
as n
tends to infinity. 
A
sequence {a_{n}}
is
convergent if it has a limit. Otherwise, we
say the sequence is divergent. 
Thus,
the sequences, (3),
(4)
and (5),
from the example above, all converge or tend to the limit 1. 

Example: Let
examine the limit of the sequence (3)
given by 



Solution:
Prove that 



The
sequence 

tends
to the limit 1
as the distance 

can
become 

arbitrary small by
choosing the natural number n
sufficiently large. 
For
example, if we choose
n = 100
the distance between a_{n}
and 1
is 1/n
= 0.01
that is, 
for
all n > 100
the distance  a_{n}

1  < 0.01. 
Therefore,
n_{0}(e)
= 101 meaning, starting from the 101^{st}
term further, the distance of the remaining terms of the
sequence and 1,
is always less than 0.01. 










Calculus
contents B 



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