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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?

Sebastiano
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1 Answers1

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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 ($[(-\pi/2)/2\pi, (\pi/2)/2\pi] \subseteq (-1,1)$).

Eric Towers
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  • Thank you Eric for your answer. I wanted to know only the upper bounds (with the natural exponent $n$) that I have written with the color red. Can you edit your answer, please? – Sebastiano Mar 26 '24 at 21:54
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    @Sebastiano : I can't make sense of the phrase "the upper bond of each series". The value of the series is known; it's a constant; the upper bound is the lower bound is the value of the series. – Eric Towers Mar 27 '24 at 06:48
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    @Sebastiano : Perhaps review of https://math.stackexchange.com/questions/947524/rule-for-the-power-of-absolute-value-expressions would be of use to you? – Eric Towers Mar 27 '24 at 06:55
  • Eric see only if is true the red signs. – Sebastiano Mar 27 '24 at 08:56
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    @Sebastiano Eric's link seems to be saying that you have = at the red signs. If you're asking about something else, then I'm having trouble understanding what you mean. – Teepeemm Mar 27 '24 at 13:51
  • @Teepeemm Thank you very much to a user also of TeX.SE. :-) – Sebastiano Mar 27 '24 at 21:18