Sequences And Series

       
           

In mathematics the ordered list of objects is called as sequence. Like sets it also consists of elements and the number of ordered elements which can be infinite and it is known as the length of sequence. Sequence is a discrete function in mathematics. Suppose we are given with a sequence (C, E, Z) but (E, Z, C) is not a sequence because it does not define order and in sequence order matters. Sequence is of two types i.e. finite or infinite sequence. The above example is of finite sequence and for example of infinite sequence we can consider all whole numbers (1, 2, 3, 4…). Therefore infinite sequence sometimes known as streams and finite number are known as strings or words. There are two main categories of sequence these are arithmetic sequence and geometric sequence. Arithmetic sequence is the sequence where each term is constant which is greater or less than the previous term, like in these types of sequence an+1 = an + d where ‘d’ is constant. The geometric sequence is a sequence in which the proportionate ratio between each term and the previous term appears to be constant.

A subsequence is the type of sequence which is formed by editing some of the elements of the sequence even without disturbing their positions and also by not hampering the positions of the remaining elements. There are also many other possible sequence, suppose if the terms of sequence are integers then the sequence is integer sequence. If the term of the sequence is polynomial then the sequence is known as polynomial sequence.

When the terms of sequences and series are added that becomes the series it also has addition operation between them. Series is considered as the important part of calculus and it starts with the sequence. Moreover if the (a1 ,a2 ,a3 ,....) then one can even consider the sequence of partial sums as (S1 , S2 , S3 , …) with:

 Sn = a1 + a2 + a3 +....+ an = ∑ni=1 xi

 Also the finite sequence and series has the defined first term and the defined last term, whereas infinite series are infinite. These terms is generated with the help of formula and algorithm. There exists infinite number of terms therefore sometimes this notion is known as infinite series. 

This is all about series and sequences.

Math Topics