I was thinking something like this - in a Fourier series the period of the $n$th term where $p$ is the period of the function being considered would be at $ x = \dfrac{2p}{n}$ . If we were to take the lcms of all these time periods, we'd get to the point where the function would repeat again. Which would be $p$
So Shouldn't the limit as $n$ goes to infinity infinity of $ \dfrac{(2p)^n}{n!} $ equal $p$?
The LCM of $24$ and $36$ is $72,$ but $24\times 36 = 864.$ Simply multiplying doesn't generally find the LCM. Especially when you have a pattern such as that in $2p,\ 2p/2,\ 2p/3,\ 2p/4,\ 2p/5,\ \ldots$ In that case, $2p$ is the LCM.