In Calculus we teach that if the $a_n$ are positive and decreasing with limit equal to zero, then the alternating series $\sum_n (-1)^na_n$ converges. One can in general not leave out the assumption that the $a_n$ (eventually) decrease, as the example $a_{2n}:=1/n$ and $a_{2n+1}:=1/2^n$ shows. However, most examples are series where the $a_n$ are given by some function $a_n=f(n)$ (for $n\gg 0$).

So my question is, for which class of functions $\mathcal F$ do we have the property that if $f$ is a positive function in $\mathcal F$ and $\lim_{x\to \infty}f(x)=0$, then the alternating series $\sum_n (-1)^nf(n)$ converges. For instance, any o-minimal class will work, since any $f$ with limit zero at infinity must eventually be decreasing. But I think if we add the sine function to this class and close under addition, multiplication and composition, this is still true. In fact, I would almost dare to postulate that the class of elementary functions (i.e., the ones our students work with), have this property, and so we do not need to ``bug'' them with this extra condition.

anyfunction and modify it to have $f(\frac1n)=0$ for all $n\gg0$. I think that you might want to put some more restrictions, or elaborate slightly more on $\cal F$. (Also double spaces break into paragraphs and ease up the reading.) – Asaf Karagila Feb 20 '13 at 23:18