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Divergence Test

If the series converges, then the sequence converges to zero. Equivalently:

If the sequence does not converge to zero, then the series can not converge.

This test can never be used to show that a series converges. It can only be used to show that a series diverges. Hence, the second version of this theorem is the more important, applicable statement.

Examples 4.2.2:
 
  • Does the Divergence test apply to show that the series converges or diverges ? How about convergence or divergence of the series ?
Proof:

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.

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