Real Analysis: Alternating Series Test (2024)

4.2. Convergence Tests

Alternating Series Test

This test does not prove absolute convergence. In fact, when checking for absolute convergence the term 'alternating series' is meaningless.

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.