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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.

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