Cauchy's Convergence Criterion
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Theorem
Cauchy's Convergence Criterion on Real Numbers
Let $\sequence {x_n}$ be a sequence in $\R$.
Then $\sequence {x_n}$ is a Cauchy sequence if and only if $\sequence {x_n}$ is convergent.
Cauchy's Convergence Criterion on Complex Numbers
Let $\sequence {z_n}$ be a complex sequence.
Then $\sequence {z_n}$ is a Cauchy sequence if and only if it is convergent.
General Case
Let $\sequence {x_n}$ be a sequence in $\R$ or $\C$.
Then $\sequence {x_n}$ is a Cauchy sequence if and only if:
- $\forall \epsilon \in \R_{>0}: \exists N \in \N: \forall r \in \N: r \ge N: \forall k > 0: \size {\ds \sum_{i \mathop = 1}^k x_{r + i} } < \epsilon$
Also known as
Cauchy's Convergence Criterion is also known as the Cauchy Convergence Condition.
Source of Name
This entry was named for Augustin Louis Cauchy.
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Cauchy convergence condition: 1.
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Cauchy convergence condition: 1.