Cauchy Sequence

Definition

A Cauchy sequence is a sequence where the elements become arbitrarily close to each other as the sequence progresses.

Examples

Cauchy Sequence

${\Sigma }_{n=1}^{\infty }\frac{1}{n^2}=1,\frac{1}{4},\frac{1}{9},\dots$
${\lim }_{n\to \infty }\frac{1}{n^2}=0$
Which this sequence converges to 0, towards infinity

Non-Cauchy Sequence

${\Sigma }_{n=1}^{\infty }(-1{)}^n=-1,1,-1,1,\dots$
These never converge to a limit, hence it is not Cauchy.

Furthermore, here, using something like ${\lim }_{n\to \infty }(-1{)}^n$ is nearly impossible to know what the value would be as $\infty$ is neither even or odd.