4.2. Convergence Tests
If the sequence does not converge to zero, then the series can not converge.
Suppose the series does converge. Then it must satisfy the Cauchy criterion. In other words, given any > 0 there exists a positive integer N such that whenever n > m > N then
| | <Let m > N and set n = m. Then the series above reduces to
| a n | <if n > N. That, however, is saying that the sequence converges to zero.