LIMIT OF A SEQUENCE Meaning and
Definition
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The limit of a sequence refers to the value that the terms of the sequence tend to approach as the index approaches infinity. In mathematics, a sequence is a list of numbers in a specific order. The limit represents the ultimate destination or target value towards which the sequence converges.
Formally, the concept of a limit of a sequence can be defined as follows. Let (a_n) be a sequence, where n is the index representing each term. A number L is said to be the limit of the sequence (a_n) if, for any positive error margin ε, there exists a positive integer N such that for all indices n greater than or equal to N, the terms of the sequence a_n are within ε distance from L.
In simpler terms, this definition implies that as the index of the terms increases towards infinity, the values of the terms eventually get arbitrarily close to the limit L. However, it is important to note that the limit need not actually be a term in the sequence itself.
The concept of a limit of a sequence is fundamental in analysis and has various applications in calculus and other branches of mathematics. It allows mathematicians to study the behavior and convergence properties of sequences, providing insights into their long-term trends and potential outcomes.
