# Thread: Cauchy sequences and series

1. ## Cauchy sequences and series

What is the relationship between the Cauchy Criterion for sequences and Cauchy Criterion for series?

2. Originally Posted by izzydoesit
What is the relationship between the Cauchy Criterion for sequences and Cauchy Criterion for series?
The Cauchy Criterion for a sequence states that for EVERY epsilon > 0, there exists an N a natural number such than for n,m > N

$\displaystyle |a_n-a_m|< \epsilon$

The Key thing to remember is that a series is a sequence of partial sums

The Cauchy Crterion for a series states that for EVERY epsilon > 0, there exists an N a natural number such than for n,m > N

$\displaystyle | \sum_{i=n}^{m}a_i| < \epsilon$

but what is this exactly? Well

$\displaystyle S_n=a_0+a_1+...a_n$
$\displaystyle S_m=a_0+a_1+...a_n+a_{n+1}+...a_m$

These are both members of the sequence of partial sums

$\displaystyle |S_m-S_n|=|a_{n+1}+...a_m|=|\sum_{i=n}^{m}a_i|$

Now since we are talking about a sequence(of partial sums) we can use the Cauchy Criterion for sequences.

I hope this clears it up.

Good luck.