

The limit of a sequence
theorems 
The
cluster point or accumulation point 
Divergent
sequences 
Sufficient condition for convergence of a sequence 
The Cauchy criterion (general principle of convergence) 






The
cluster point or accumulation point 
Every
sequence of numbers does not have to tend to a limit value. For example
the terms of the sequence, 

shown
on the number line, 

accumulate around two points, the odd terms
around 0 while the even terms accumulate around 2. Therefore, it
is the divergent sequence. 
The
cluster point is such a point of a sequence in every however small
neighborhood of which lie infinitely many
terms of the sequence. In the example above such points are 0 and 2. 
According
to this definition every limit point is the cluster point but inverse
does not hold. 
That
is, a convergent sequence can have at most one cluster point, hence if a
sequence converges, then its limit
is unique. 

Divergent
sequences 
A
sequence is convergent if it has a limit, otherwise it is divergent. A
divergent sequence has no a finite limit. 
The
sequence of the natural numbers 1,
2, 3, . . .
, n, . . .
has
no a cluster point since it tends (or diverge) to
infinity,
so we write 

Generally, if
terms of a sequence {a_{n}}
can become greater than every arbitrary large natural number N, that is,
if for every given number N
there exists an index n_{0}
such that 
a_{n}
> N
for all n >
n_{0}(N), 
we
say that the sequence {a_{n}}
tends to infinity or diverge to +
oo
and we write either 
a_{n}
®
+ oo
as n
®
oo 
or 



If
terms of a sequence {a_{n}}
are such that for every given negative number 
N,
of arbitrary large absolute value,
there exists an index n_{0}
such that 
a_{n}
> 
N
for all n >
n_{0}(N), 
we
say that the sequence {a_{n}}
tends to the negative infinity or diverge to 
oo
and we write either 
a_{n}
®

oo
as n
® oo 
or 



For
example, the
sequence {
n}
tends to negative infinity or diverge to 
oo
. 

Sufficient condition for convergence of a sequence
 The Cauchy criterion (general principle of convergence) 
A
sequence of real numbers, a_{1},
a_{2},
. . . ,
a_{n},
. . .
will have a finite limit value or
will be convergent if for no matter
how small a positive number e
we take there exists a term a_{n}
such that the distance between that term
and every term further in the sequence is smaller than e,
that is, by moving further in the sequence the
difference between any two terms gets
smaller and smaller. 
As
a_{n + r},
where
r = 1, 2, 3,
. . . denotes any term that follows a_{n},
then 

a_{n + r} 
a_{n}
 < e
for all n >
n_{0}(e),
r
= 1, 2, 3,
. . . 
shows
the condition for the convergence of a sequence. 
If
a sequence {a_{n}}
of real numbers (or points on the real line) the distances between which
tend to zero as their
indices tend to infinity, then {a_{n}}
is a Cauchy sequence. 
Therefore,
if a sequence {a_{n}}
is convergent, then {a_{n}}
is a Cauchy sequence. 

The Cauchy criterion
or general principle of convergence,
example 
The
following example shows us the nature of that condition. 
Example: We
know that the sequence 0.3,
0.33,
0.333,
. . . converges to the number 1/3
as 
1/3
= 0.33333
. . . . Let
write the rule for the n^{th}
term, 

If we go along the sequence far enough, say to the 100^{th} term, i.e., the
term with a hundred 3's in the fractional
part, then the difference between that term and every next term is equal
to the decimal fraction with the
fractional part that consists of a hundred 0's followed by 3's on the
lower decimal places, starting from the
101^{st} decimal place. That is, 

Therefore,
the absolute value of the difference falls under 


Then,
if we go further along the sequence and for example calculate the
distance between the 100000^{th} term 
and the following
terms, the distance will be smaller than 


Hence,
since we can make the left side of
the inequality 
a_{n + r} 
a_{n}
 < e as
small as we wish by choosing n
large enough, then all terms that follow a_{n}
(denoted a_{n + r},
r = 1, 2, 3,
. . . ), infinitely many of
them, lie in
the interval
of the length 2e symmetrically
around the point a_{n}.
Outside of that interval there is only
a finite number of terms. That is, 

e <
a_{n + r} 
a_{n}
< + e for all
n >
n_{0}(e),
r
= 1, 2, 3,
. . . 
or a_{n}

e <
a_{n + r} < a_{n}
+ e. 
So,
the
terms of the sequence, starting from the (n +
1)^{th} term, form the
infinite and bounded sequence of numbers
and so, according to the above
theorem, they must have at least one cluster point that lies in that
interval. But they cannot have more than one cluster point since all
points that follow the n^{th} term lie inside the interval
2e length of
which is arbitrary small, if n
is already large enough, so that any
other cluster point will have to be outside of that interval. 
Thus,
the
above theorem simply says that if a sequence converges, then the terms of the sequence
are getting closer and closer to each
other as shows the example. 











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