# 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.