@J.M. I really appreciate you taking the time to share your knowledge in this thread. But I think I'd like to backtrack a little bit here.
I think for my purposes, this equation that you provided is most useful.
$J_n(x)\approx\frac1{m}\sum_{k=0}^{m-1}\sin\left(x\sin\left(\frac{\pi}{2m}\left(k+\frac12\right)\right)\right)\sin\left(\frac{\pi n}{2m}\left(k+\frac12\right)\right)$
At the end of the day, I think what I need to know is how $\beta$ affects the $J_o(\beta)$. As you probably know, $\beta$ (in communication theory) is equal to $\frac{\Delta \omega}{\omega_m}$ such that $\Delta \omega$ is constant and $\omega_m$ is the modulation frequency. For my case, I have also defined $s = j\omega_m$. (This is important for later)
I have also taken your suggestion and used m = 8. Therefore, we can simplify the above equation to be
$J_n(\frac{\Delta \omega}{\omega_m})\approx\frac1{8}\sum_{k=0}^{7}\sin\left(\frac{\Delta \omega}{\omega_m}\sin\left(\frac{\pi}{16}\left(k+\frac12\right)\right)\right)\sin\left(\frac{\pi n}{16}\left(k+\frac12\right)\right)$
Essentially, since I will be doing a frequency response later, I want to change the $\omega_m$ term to be in terms of $s$. So, we can do the substitution for $\omega_m = -js$. So if, we also choose n = 2 (as an example, we are left with.
$J_2(\frac{\Delta \omega}{\omega_m})\approx\frac1{8}\sum_{k=0}^{7}\sin\left(\frac{\Delta \omega}{-js}\sin\left(\frac{\pi}{16}\left(k+\frac12\right)\right)\right)\sin\left(\frac{2\pi }{16}\left(k+\frac12\right)\right)$
The problem I am seeing is that for large values of $\Delta\omega$ (around 4k for this case), and putting that into a cosine, $J_n(\beta)$ returns infinite.
Is there something obvious that I am missing here? It doesn't make sense that I cannot evaluate for this size of $\beta$.
I also see in your most recent post that this equation
$|J_n(x)|\leq\frac1{n!}\left|\frac{x}{2}\right|^n$
This actually looks much simpler to use and since $\beta$ is positive, I can ignore the absolute values on the right hand side, and would also not have to deal with the cosine/sines like above. Do you think this is possible?
Sorry in advance for the long read