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.