## 4.2. Convergence Tests

### Alternating Series Test

A series of the form
with

This test does not prove absolute convergence. In fact, when checking
for absolute convergence the term 'alternating series' is meaningless.
*b*is called_{n}0**Alternating Series**. If the sequence is decreasing and converges to zero, then the sum converges.It is important that the series truly alternates, that is each positive term is followed by a negative one, and visa versa. If that is not the case, the alternating series test does not apply (while Abel's Test may still work).

**Proof:**

Let *a _{n} = (-1) ^{n}*. Then the formal sum
has bounded partial sums (although the sum does not itself converge.
Why not ?). Then, with the given choice of
Abel's test applies directly, showing that the series indeed converges.