sequence and series

                                                      

  Sequence and series 

   Sequence and series are the basic topics in Arithmetic. An itemized collection of elements in which repetitions of any sort are allowed is known as a sequence, whereas a series is the sum of all elements. An arithmetic progressions is one of the common examples of sequence and series.

• In short, a sequence is a list of items/objects which have been arranged in a sequential way.
• A series can be highly generalized as the sum of all the terms in a sequence. However, there has to be a definite relationship between all the terms of the sequence.

Sequences are fundamental to the study of infinite series and many applications of mathematics.

 Representing Sequences

 A sequence is a list of numbers 

                      a1 , a2 , a3 , . . . , an , . . .

in a given order. Each of a1, a2, a3 and so on represents a number. These are the terms of the sequence. For example, the sequence 2, 4, 6, 8, 10, 12, c, 2n, c has first term a1 = 2, second term a2 = 4, and nth term an = 2n. The integer n is called the index of an, and indicates where an occurs in the list. Order is important. 

Kinds of Sequences 

 1. Finite Sequence: A sequence  < an > in which  an=0  ∀ n> m E n is said to be a finite Sequence. i.e., A finite Sequence has a finite number of terms.

 2. Infinite Sequence: A sequence, which is not finite, is an infinite sequence.

convergence and divergence of sequence 

It is important to remember that $\sum _{i=1}^{\infty }{a}_i$  is really nothing more than a convenient notation for $\underset{n\to \infty }{lim}\sum _{i=1}^n{a}_i$ so we do not need to keep writing the limit down. We do, however, always need to remind ourselves that we really do have a limit there!

If the sequence of partial sums is a convergent sequence (i.e. its limit exists and is finite) then the series is also called convergent and in this case if $\underset{n\to \infty }{lim}{s}_n=s$ then, $\sum _{i=1}^{\infty }{a}_i=s$. Likewise, if the sequence of partial sums is a divergent sequence (i.e. its limit doesn’t exist or is plus or minus infinity) then the series is also called divergent

                            

Let’s take a look at some series and see if we can determine if they are convergent or divergent and see if we can determine the value of any convergent series we find.

EXAMPLE 1.

 check convergence and divergence of (5n-1)/(4n-1).

SOLUTION 

for checking  convergence or divergence of a sequence we have to check limit of the sequence at infinity ,if limit exist it converge if not then it diverges

 lim   (5n-1)/(4n-1)          = lim  n(5-1\n)/n(4-1/n) 

  n→∞                                     n→∞                                          

                                                   =5/4

so limit of the sequence at infinity is a constant, Hence the given series is convergence series.

EXAMPLE 2

 check convergence and divergence of (n^2-2n+1)/(n-1).

 SOLUTION 

for checking  convergence or divergence of a sequence we have to check limit of the sequence at infinity ,if limit exist it converge if not then it diverges

 lim   (n^2-2n+1)/(n-1)   = lim  (n-1)^2/(n-1) 

           n→∞                                     n→∞     

                                                 =lim (n-1)

                                                               n→∞                  

                                                 =∞                                

so limit of the sequence at infinity is a infinity, Hence the given series is divergence series.

the following operations can be done on a sequence