Can an alternating series ever be absolutely convergent?

Hint: You could take any (absolutely) convergent series $\sum_{n=0}^\infty{a_n}$ where $a_n> 0$, and then consider $\sum_{n=0}^\infty{(-1)^na_n}$.

$$ \sum_{n=0}^\infty \frac{(-1)^n}{2^n} = \frac 2 3. \qquad \sum_{n=0}^\infty \frac 1 {2^n} = 2. $$

a series is absolutely convergent if $\sum |a_n| < M$

If a series is absolutely convergent then every sub-series is convergent.

Consider $\sum (-1)^n|a_n|$ The sum of the of the even terms converges, the sum of the odd terms converges.