Sequences and limits
      Infinite sequences
         Sequences notation - the rule for the n-th 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
         Verifying the convergence of a sequence from the definition, examples
Infinite Sequences
An infinite sequence is an ordered list of real numbers indexed by the natural numbers n N denoted  {an}
a1a2a3, . . . ,  an, . . .
where by an given is a rule to calculate the nth term of the sequence.
Graphing the terms of a sequence on the number line
Thus, for example;
 (1)              an = 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 {an} if for every positive number e there exist a natural number 
n0  such that if  n > n0,  then   | an - L | < e.
That is, a number L is the limit of a sequence if the distance between the term an and L  becomes arbitrary small by choosing n large enough (see the examples above). The n0 denotes the value of the index n starting from which the distance  an - L  becomes smaller than the given e.
Since the value of n0 depends on the size of e it is usually written as n0(e).
Convergence of a sequence
Therefore, if a sequence {an} has a limit L we write
  or  an L    as   n oo  
and we say that that a sequence an has the limit L as n tends to infinity.
A sequence {an} 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 an and 1 is 1/n = 0.01 that is,
for all  n > 100  the distance  | an - 1 | < 0.01. 
Therefore,  n0(e) = 101 meaning, starting from the 101st term further, the distance of the remaining terms of the sequence and 1, is always less than 0.01.
Verifying the convergence of a sequence from the definition examples
Example:  Find the limit of the sequence (2) given by  
Solution:   Let prove that   
Using the definition of limit we must find a natural number n0(e) such that  | an - 0 | < e  for all  n > n0.
Therefore, if n > n0  then   |1/n - 0 | = |1/n | = 1/n < 1/n0 < e.
Suppose we wish to make the difference (or the distance) between the anth term and the limit L to be less than e = 0.001 = 1/1000. 
Then, as  1/n < e  or  n > 1/e  it follows  n > 1000  that is, starting from  n0 = 1001  the distance  an - L becomes smaller than the given e.
Observe that the absolute value of terms of the sequences that converge to zero become arbitrary small as n tends to infinity that is,       | an | < e  for all  n > n0(e). 
For example such sequences are,   therefore  
The same way we can prove that the sequence (5) above, given by converges to 1. 
As can be seen on the number line above, the terms of the sequence alternate from left to right approaching closer and closer to 1 as n tends to infinity.
That is, the sequence   alone alternately converges from left to right approaching closer and
closer to 0 as n tends to infinity.
Example:  Find the limit of the sequence,    0.9,   0.99,   0.999, . . . ,  0.999  . . . 9 . . . , . . .
Solution:   The terms of the sequence can be written as
Therefore,  
Example:  The sequence has the limit 3/2, starting from which term  | an - L | < 0.01.  
Solution:  Substitute given values into the inequality | an - L | < 0.01,
   
Check the result by plugging  n0(e) = 125  into  | an - L | < 0.01.
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