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A Convergent Sequence is a sequence which becomes arbitrarily close to a specific value, called its "limit". The sequence may or may not take the value of the limit.

The formal definition of a sequence, $\langle x_n\rangle$ , tending to a limit $L$ is:

$\forall\varepsilon>0,\exists N\in\N$ such that $\forall n\ge N,|x_n-L|<\epsilon$ .

It may be written $\lim_{n\to\infty}x_n=L$ , or $x_n\xrightarrow[n\to\infty]{}L$ .

Proving that a sequence converges from the definition requires knowledge of what the limit is. Therefore we introduce the notion of Cauchy sequences, which is an equivalent condition for convergence (by the General Principle of Convergence).

Examples

$n$ to the power of 0 (starting with $n=1$) converges to 1.

$n^0\to1$

The $n$-th root of $x>0$ converges to 1.

$\sqrt[n]{x}\to1$