## 4.2. Convergence Tests

### Divergence Test

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

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

*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.

**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

if| a_{n}| <

*n > N*. That, however, is saying that the sequence converges to zero.