My textbook gives this definition of the alternating series test:
Test for alternating series. An alternating series conerges if the absolute value of the terms decreases steadily to zero, that is, if $|a_{n_1}| \le |a_n|$ and $\lim_{n\to\infty}a_n = 0$.
Similarly, Wikipedia gives this definition:
...the alternating series test tells us that an alternating series will converge if the terms $a_n$ converge to 0 monotonically.
I don't see an "if and only if" in either definition. My question is: If the terms $a_n$ do not converge to 0 monotonically, do I know that the series diverges? Or is the test inconclusive?
If the terms do not converge to $0$, then we cannot have convergence. But if monotonicity is abandoned, then we could have convergence or divergence.
For instance, the series $$-\frac{1}{2^2}+\frac{1}{1^2}-\frac{1}{4^2}+\frac{1}{3^2}-\frac{1}{6^2}+\cdots$$ converges, indeed converges absolutely.
The series $$-\frac{1}{2}+\frac{1}{1}-\frac{1}{4}+\frac{1}{3}-\frac{1}{6}+\cdots$$ also converges.
We now give an example of divergence. Write down the usual harmonic series $1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\cdots$. In between all these positive terms, put in, in turn, $-\frac{1}{2}$, $-\frac{1}{2^2}$, $-\frac{1}{2^3}$, $-\frac{1}{2^4}$ and so on.
One can produce more interesting examples. Take the familiar alternating series $1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\cdots$.
We can rearrange the terms of this series, so that they still alternate in sign, but the resulting series diverges.