## 4.1. Series and Convergence

### Theorem 4.1.9: Cauchy Criteria for Series

The series
converges if and only if for every

*> 0*there is a positive integer*N*such that if*m > n > N*then*| | <*### Proof:

Suppose that the Cauchy criterion holds. Pick any*> 0*. Then

But that means precisely that the sequence of partial sums| S_{n}- S_{m}| = || <

*{ S*is a Cauchy sequence, and hence convergent.

_{N}}
Now suppose that the sum converges. Then, by definition, the
sequence of partial sums converges. In particular, that sequence
must be a Cauchy sequence: given any
* > 0*, there is
positive integer *N* such that whenever
*n, m > N* we have that

But that, in turn, means that the Cauchy criterion for series holds.| S_{n}-S_{m}| = | | <