cauchy sequence

**flower3** · December 28th 2009, 08:11 AM

Prove that the sequence :
$x_n = n^2 + \frac{(-1)^n}{n}$ is not cauchy

**tonio** · December 28th 2009, 08:37 AM

Originally Posted by flower3

Prove that the sequence :
$x_n = n^2 + \frac{(-1)^n}{n}$ is not cauchy

If it were Cauchy then for any $\epsilon >0\,\,\,\exists M\in\mathbb{N}\,\,\,s.t.\,\,\,|x_n-x_m|<\epsilon\,\,\,\forall\,n,m>M$ . In particular, this must be true for $m=n+1$ , but:

$|x_n-x_{n+1}|=\left|-2n-1+\frac{(-1)^n}{n(n+1)}\right|=2n+1\pm \frac{1}{n(n+1)}>2$ , so it is enough to take $\epsilon < 2$ and the above definitory property of Cauchy sequences won't be true for it.

Tonio

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