Limit and Alternation: Understanding the Alternating Series Test

[mohabitar]

The alternating series test states that if b[n] is decreasing and the lim n-->infinity of b[n]=0, then the series converges. However, does this also mean that if one of the two conditions is not satisfied, then the series is automatically divergent?

 

[Mark44]

The test also requires the terms of the series to be nonnegative, where the alternation is taken care of by (-1)ⁿ, as in
$$\sum_{n = 0}^{\infty} (-1)^n b_n$$

In general, if all of the conditions of a test aren't met, it means that you can't use that test. In the case of this test, you could have a series where b_n is not decreasing, but the series could still converge. For example, if b_n $\equiv$ 0. (Since you ignored this condition, I will, too.)