generated from smyalygames/quartz
13 lines
654 B
Markdown
13 lines
654 B
Markdown
# Definition
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A **Cauchy sequence** is a sequence where the elements become arbitrarily close to each other as the sequence progresses.
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# Examples
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## Cauchy Sequence
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$$\Sigma_{n=1}^\infty \frac{1}{n^2} = 1, \, \frac{1}{4}, \, \frac{1}{9}, \, \dots$$
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$$\lim_{ n \to \infty } \frac{1}{n^2} = 0 $$
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Which this sequence converges to 0, towards infinity
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## Non-Cauchy Sequence
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$$\Sigma_{n=1}^{\infty}(-1)^n = -1, \, 1, \, -1, \, 1, \, \dots$$
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These never converge to a limit, hence it is not Cauchy.
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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.
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