generated from smyalygames/quartz
654 B
654 B
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.