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As the title indicates, I am trying to find a Cauchy sequence that is not fast (or rapidly) Cauchy. Could anyone suggest something?

A sequence $\{a_n\}_{n \in \Bbb N}$is termed fast (or rapidly) Cauchy if there is a convergent series or positive numbers $\sum_{k \in \Bbb N} \epsilon_k^2$ for which

$$\|a_k - a_{k+1}\| \le \epsilon_k^2 \ \forall k $$

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what do you mean by fast Cauchy? – Chris Eagle Feb 28 '13 at 0:40
Updated the question. I thought the term was standard that's why I did not include it the first time. – user44069 Feb 28 '13 at 0:46
I know that the meaning is obvious, but you talk about $\epsilon_k^2$ and $e_k^2$. Besides, the exponent $2$ is unnecessary. – Matemáticos Chibchas Feb 28 '13 at 0:52
up vote 7 down vote accepted

$\dfrac{(-1)^n}{n}$ is convergent to $0$ (and hence Cauchy) but is not fast Cauchy.

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