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Limit of a sequence

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Given an infinite sequence \{x_n\} whose values are real numbers (or otherwise members of a common metric space), the limit of the sequence is a value L which the values of the sequence approach, if such a value exists.

That is, we say L=\lim_{n\to\infty}x_n , if for every \varepsilon>0 , there exists a N\in\N such that k>N implies |x_k-L|<\varepsilon (or d(x_k,L)<\varepsilon in an arbitrary metric space).

That is, all values of the sequence after some N-th term are within \varepsilon of the limit value. Put simply, this means that you can choose any "maximum distance" from an element of the sequence to its limit, and there will only be a finite number of elements that do not fall into this range, no matter how small the distance chosen.

The most common method of evaluating the limit of a sequence is to make a function f(n)=a_n , and taking the limit at infinity. For example,


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