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I noticed that the sum of certain functions composed prime-number of times sometimes converges and I'm curious what we can say about the properties of functions that do converge and those that do not. To further explain...

Let $f_n(\alpha)$ be defined as the function $g$ composed $n$ times evaluated at $\alpha$. $$ f_n(\alpha) = (g \circ \overbrace{ \ldots }^n \circ g)(\alpha), \,\alpha \in \mathbb{R}$$

Now consider the function

$$ s(\alpha) = \sum_{n=1}^{\infty} (-1)^{n-1} f_{P_n}(\alpha) $$ where $P_n$ is the n-th prime number and $ g(\alpha) = \sin(\alpha) $. The partial sums appear to be approaching $0.3695$ ($\approx 30 e^{\Gamma(5/24)}$). Are the top and bottom bounded or do they both continue to approach $y=0.3695$? Below is a plot of the partial sums: partial sums of (s)

Another interesting plot is when $g(\alpha) = \ln|\alpha + 1| $. Here the series seems to certainly converge to $0.3704$ partial sum of log

Is there a closed form solution when $g(\alpha) = \ln|\alpha +1|$? The value of convergence appears to depend on $\alpha$ so I propose there might be a closed form solution in terms of $\alpha$. I've observed that as $\alpha$ increases so does the value of convergence.

Are there any other statements we can make about the convergence of $s(\alpha)$ given the characteristics of $g$? Given $g$, can we determine what values of $\alpha$ cause $s(\alpha)$ to converge? Or if it converges at all?

Update 1: A table of the values of $s(\alpha)$ when $g(\alpha)=\ln|\alpha +1|$:

s(0)  = 0.000000
s(1)  = 0.252109
s(2)  = 0.370438
s(3)  = 0.445306
s(5)  = 0.540452
s(10) = 0.664692
s(50) = 0.913516

Update 2: More examples of functions that appear to cause $s$ to diverge are $g(\alpha)=\frac{\tan(\alpha)}{\alpha}$, $g(\alpha)=\tan(\alpha + \sin(\alpha))$, $g(\alpha)=\sin(\mathrm{e}^\alpha + \alpha)$, and $g(\alpha) = \lfloor \sqrt{\lfloor \alpha \rfloor + 1} + \alpha \rfloor$. What about these functions causes $s(\alpha)$ to diverge?

Update 3: Below is a plot of $g(\alpha) = \lfloor \sqrt{\lfloor \alpha \rfloor + 1} + \alpha \rfloor$

plot of g

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    I've discussed related questions giving it (and proposing everywhere) the standard name: "iteration series", and if having alternating sign "alternating iteration series" ("AIS"). I've looked at such alternating series in context of *tetration* (iterated exponentiation) finding some interesting properties, of course depending on the existence of finite fixpoints and so on. I even could find a method to convert AIS into powerseries (using Carleman matrices) However I've nothing using the prime-indexed AIS only. (...) – 2017-04-15
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    (...) Perhaps you might search the tetration-forum at http://math.eretrandre.org/tetrationforum/ for "iteration series" and my name, and I think there has been some stuff here in MSE and MO in the same way.(I can come back to this only later today) – 2017-04-15
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    (...) For instance you might be interested in this main thread http://math.eretrandre.org/tetrationforum/showthread.php?tid=760 – 2017-04-15
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    @GottfriedHelms thanks for he links. So what I'm getting from a glance over those links is that tetration AIS sometimes converge? – 2017-04-15
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    Yes. Consider $f(x) = b^x $ where $b=\sqrt 2$. Then beginning at $x_0=3$ iterating with $f°^h(x_0)$ this iterations converge to the fixpoint $2$. Alternatingly summing must be done by Cesaro-sum or Euler-sum or similar methods. Of course one can also look at $f(x) =b^x-1$ and use some small base where iterating converges to zero, and then one do not even need Cesaro-/Euler-summation for the alternating series because $f°^\infty(x_0)$ approaches zero itself. Another interesting AIS is that of the $f(x)=\sinh(x)$ when starting at small values since again $f°^\infty(x_0)$ approaches zero. – 2017-04-15

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