Determine if Sequence is Cauchy

Question

Can someone please tell me how to determine if a sequence is Cauchy without using the limit. I'm supposed to use partial fraction decomposition

$a_n=\frac{1}{n(n+1)}$. When I did the partial decomposition I found $\frac{1}{n}+\frac{1}{n+1}$ but I don't know what to do from there.

Thank you

I think the partial decomposition should be $\frac{1}{n}-\frac{1}{n+1}$ which is telescoping!!\ — user614287, Apr 19 '19 at 00:14
You should use MathJax. Posts that do not use MathJax tend to get less attention. — parsiad, Apr 19 '19 at 00:14
You seem to have two typos with the parenthesis, can you please edit your post and correct them? And don't forget to use MathJax to make things easier to read. — Ertxiem - reinstate Monica, Apr 19 '19 at 00:16

score 0 · Answer 1 · answered Apr 19 '19 at 00:42

0

$|a_n| <\frac 1 n$ so $|a_n-a_m|\leq |a_n|+|a_m| <\frac 1 n+\frac 1 m<\epsilon$ if $n,m >\frac 2 {\epsilon}$.

answered Apr 19 '19 at 00:42

Kavi Rama Murthy

1 Answers1