Originally Posted by Kharakov
2) Are there an infinite amount of (non trivial) alternating infinite series in the following form, with r being a variable ratio, and a being a set sequence of numbers (like 2!,4!,6!... or 1!,3!,5!... or 1,20,120, 1200... etc.)

$\lim_{t\to\infty} \sum (-1)^{t+1} \,\, \frac{r^t}{a_t} \,\, = \,\, \frac{r}{a_1} -\frac{r^2}{a_2} +\frac{r^3}{a_3} -\frac{r^4}{a_4} +...$

.. that have an infinite amount of zeros as r is increased, similar to sine and cosine?
Anyone? Are there any non-trivial examples of alternating series for periodic functions other than ones that are based on sine and cosine?

Trivial would be something like the expanded series for cos(a) * cos(b)...