Just a only doubt. Supposing that I have these three series:
$$\sum_{n=1}^\infty\left[\arcsin(p(x))\right]^n, \quad \sum_{n=1}^\infty \left[\frac{1}{2\pi}\arccos(q(x))\right]^n, \quad \sum_{n=1}^\infty \left[\frac{1}{2\pi}\arctan\left(g(x)\right)\right]^n$$
Supposing that the three goniometric inverse functions are defined into your domains.
I was interested for the upper bond of each series. Look the red colours.
$$\left|\left[\arcsin(p(x))\right]^n\right| \color{red}{=}\color{red}{or\, \leq}\left|\arcsin(p(x))\right| ^n\leq \left(\frac{\pi}2\right)^n, \quad \forall n\in\Bbb N$$
$$\left|\left[\frac{1}{2\pi}\arccos(q(x))\right]^n\right| \color{red}{\leq} \left|\frac{1}{2\pi}\arccos(q(x))\right|^n \leq \left(\frac{1}{2\pi}\cdot \pi\right)^n=\left(\frac 12\right)^n, \quad \forall n\in \Bbb N$$
For the $\arctan?$ I know that the codomain of $\arctan$ is $]-\pi/2,\pi/2[$. How is the arcotangent upper bound is done in this case? Like that of the arcosine since both are odd functions?
Um... Assuming $g$, $p$, and $q$ are real-valued... Since the trig functions don't have arguments that depend on $n$, these are three geometric series: $\sum_{n=1}^\infty \left( f(x) \right)^n = \frac{f(x)}{1-f(x)}$ (when $-1 < f(x) < 1$). So the arcsine series diverges when $\arcsin p(x) \in [-\pi/2, -1] \cup [1, \pi/2]$ (i.e., when $p(x) \in [-1,\sin(-1)] \cup [\sin(1), 1]$) and otherwise converges, the arccosine series always converges ($[0/2\pi, \pi/2\pi] \subseteq (-1,1)$), and the arctangent series always converges ($[(-\pi/2)/2\pi, (\pi/2)/2\pi] \subseteq (-1,1)$).