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| S n - S m | = || <But that means precisely that the sequence of partial sums { S N } is a Cauchy sequence, and hence convergent.
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
| S n -S m | = | | <But that, in turn, means that the Cauchy criterion for series holds.