Is there an alternating series that satisfies only one of the conditions of the Alternating Series Test that nonetheless converges?

Solution 1:

Notice that ($\sum a_n$ converges) $\Longrightarrow$ ($\lim a_n=0$). The contrapositive is sometimes called the test for divergence: ($\lim a_n\ne 0$) $\Longrightarrow$ ($\sum a_n$ diverges).

So indeed, the series you are looking for does not exist.

Solution 2:

To answer the question in the title: Yes (just not the way indicated in the body).

There are plenty of examples of nonmonotonic sequences $(a_n)$ of positive numbers such that $\sum\limits_{n=0}^\infty(-1)^na_n$ converges. For example, this will holds whenever $\sum\limits_{n=0}^\infty a_n$ converges. One example would be $a_{2n}=\dfrac{1}{(n+1)^2}$, $a_{2n+1}=\dfrac{1}{(n+1)^3}$.