Infinite series | Infinite sequence | Properties of sequence | Term of the sequence | Finite |

 INFINITE SERIES:

Infinite series are of great importance in both pure and applied mathematics. They play a significant role in physics and engineering. In fact many function can be represented by infinite series.

SEQUENCES

• Definition. 

Let X be a non empty set. A function f : N→X , whose domain is the set of natural numbers, is called an infinite sequence in X. If the domain of f is the finite of numbers {1,2,3,...,n} then it is called a finite sequence. The range of a sequence may be a subset of real of complex numbers. A Sequence is called a real sequence if its range is a subset of real numbers. A sequence is called a real sequence if its range is a subset of real numbers. In what follows, only real sequences will be studied.

LET a : N→R be a sequence. The values (images) of function a at the points 1,2,3,... will  be donate by aו , aշ, aз,... instead of a a(1) , a(2) , a(3) ,... Although a sequence is a function but it is customary that the range 

aו, aշ, aз,...an,..

of a sequence a: N→R is often called a sequence. aⵏ, a2, aз. ..an,.... are called terms of a sequence. A convenient device to specify a sequence is to state a formula for its nth. the sequence

1, 1/2, 1/3, ..., 1/n

may be specified by writing its nth term an = 1/n and the sequence is symbolically written as {1/n}.

the sequence aו, aշ, aз,...an,.... is written in bracket notation as {an}.

A sequence {an} is said to converge if its nth term approaches a defined numbers as n increase without bound. Formally, a sequence {an} is said to have the limit L, if given any e＞0, there exists a positive integers N such that

|an-L|  < ε for all n ≥ N

This is equivalent to saying that all terms of sequence, when n ≥ N, belong to the open interval

] L - ɛ, L + ε [