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.
Context

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.

Interactive Real Analysis, ver. 1.9.5
(c) 1994-2007, Bert G. Wachsmuth
Page last modified: Mar 28, 2007

*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 ?