Text book Possible Error

Your textbook is correct when it says that if $b_1\geqslant b_2\geqslant b_3\geqslant\cdots$ and if $\lim_{n\to\infty}b_n=0$, then the series $\sum_{n=1}^\infty(-1)^{n-1}b_n$ converges. But this doesn't mean that otherwise it diverges.

Note that if the first condition gets replaced by$$\text{for some $N\in\mathbb N$, }b_N\geqslant b_{N+1}\geqslant b_{N+2}\geqslant\cdots$$it is still true that the series converges. And it is natural to call alternate series test to this statement.

Your understanding is wrong. First, and most importantly, the alternating series test (AST) says that if these conditions hold, then we can guarantee convergence. But that does not mean that in order to guarantee convergence, we need the conditions to hold or that if the conditions don't hold then the series is divergent.

Second, with any series, the behaviour does not depend on the first $N$ terms for any finite $N$. That is, if you were to change $a_0, \dots, a_N$ to whatever you want, it doesn't affect the convergence behaviour. This means that if the AST conditions hold for $n \ge $ some finite $N$ then you can guarantee convergence. This also works for any of the tests.

For example, the integral test normally says that if $a_n = f(n)$ and $f$ is decreasing then $\sum a_n$ converges if and only if $\int f$ converges. But, like with the AST, we just need $f$ to be decreasing on $(N,\infty)$.