The limit of a sequence theorems
      Some important limits
      Operations with limits
         Operations with limits examples
Some important limits
(1)  Let examine convergence of the sequence given by  an = | a |n
     a)  if  | a | > 1  then we can write  | a | = 1 + h,  where h  is a positive number.  
So, by the binomial theorem
If we drop all terms beginning from the third that are all positive since the binomial coefficients are natural numbers, if  n > 2  and  h > 0, the right side become smaller, so obtained is the Bernoulli's inequality 
(1 + h)n > 1 + nh,    n > 2.
When  n oo  then 1 + nh  tends to the positive infinity too, since we can make 1 + nh  greater than any 
given positive number N, if only we take   therefore will even more tend to infinity | a |n which is
greater. Thus,
  or    
    b)  if   0 < | a | < 1  then we can write  
Since b > 1 then such that  | a |n < e  whenever however small e is,  
this inequality can be satisfied by choosing n large enough. Therefore,
  or    
(2)  Let examine convergence of the sequence given by
The sequence the n-th term of which we can write as
For every | a | > 1 there exists a natural number m such that  m < | a | < m + 1  and  n > m  then
since it follows that  an  0   or we write 
   
(3)  Let examine convergence of the sequence given by  
     a)  If   a > 1  then the sequence  is decreasing, that is
 Let  then by the Bernoulli's inequality  a > 1 + nh   so that
Since the numerator  a - 1  is fixed number then, if  n  0  then  h  0  too, therefore
So we can write
     b)  If   0 < a < 1  then the sequence  is increasing, that is
For example,
If we write 
so it follows that Therefore,
     c)  If   a = 1    then 
Since in all three cases above,  a), b) and c) we've got the same result, then we can write
   
Operations with limits
We usually use following results when finding the limit of a sequence.
Let {an} and {bn} be two sequences of real numbers such that an a  and  bn b. Then,
   
   
   
   
   
   
   
Operations with limits examples
Let apply the above operations with limits to calculate limits of given sequences.
Example:  Find   
Solution: 
Example:  Find   
 
Solution:
First factor the term of highest degree from both the numerator and denominator.
Note that the same procedure can be applied to every fraction for which the numerator and denominator are polynomials in n.
The limit of such a fraction is the same as the limit of the quotient of the terms of highest degree.
Example:  Find   
Solution: 
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