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.