1. ## cauchy sequence

Show directly that the following sequence is not cauchy :
$\displaystyle x_n=ln \ n$

2. Hint:what is $\displaystyle \left|x_{2n}-x_{n}\right|$?

3. Originally Posted by flower3
Show directly that the following sequence is not cauchy :
$\displaystyle x_n=ln \ n$
It is easy to show that if $\displaystyle N\in\mathbb{Z}^+~\&~k\in\mathbb{Z}^+$ then $\displaystyle \frac{k}{N+k}\le \ln(N+k)-\ln(N)$.

4. Originally Posted by flower3
Show directly that the following sequence is not cauchy :
$\displaystyle x_n=ln \ n$
Originally Posted by Shanks
Hint:what is $\displaystyle \left|x_{2n}-x_{n}\right|$?
I think what Shanks meant was consider that for every $\displaystyle n>1$ it is obviously true that $\displaystyle n^2>n$, but $\displaystyle \left|\ln\left(n^2\right)-\ln(n)\right|=\left|2\ln(n)-\ln(n)\right|=\left|\ln(n)\right|=\ln(n)$, which is clearly greater than any $\displaystyle \varepsilon>0$ for sufficiently large $\displaystyle n$.

5. Originally Posted by flower3
Show directly that the following sequence is not cauchy :
$\displaystyle x_n=ln \ n$
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