Limits of a Sequence

The limits of a sequence is one of the important concepts of mathematics. The sequence or a series when tends to converge at a point then that point is called as the limit. Let us consider that we are having a sequence of points and a point L is said to be the limit of the sequence. In this article we are going to see about the limit of a sequence.

Definition of Limit of a Sequence:

Let us consider a series of real numbers {xn │n € N}, where the real number L is called as the limit of the sequence xn such that

lim xn = L
n->∞

which is true for every real number ε > 0 and there appears the number N such that for every number n>N,there exists │xn - L│ < ε.

Let us consider that for a series of points that has been given by { xn │n € N }with the distance function f. Then the element L € M is called as the limit of the sequence which is given by lim xn = L, which is true for every real number ε > 0 and there appears the number N such
n->∞

that for every number n>N, there exists (f(xn), L) < ε.

Let us consider an element L € T as the limit of the sequence only if for neighborhood N of L, there exists a number N when xn € N for all the values of n > N.

If a sequence contains a limit then the sequence is said to convergent and it converges up to the limit, else the sequence is said to be the divergent sequence. A null sequence does not contain any limit and it converges to zero.