<sub> Simplified repost of [Are these continued fractions of integrals known?](https://math.stackexchange.com/questions/3285230/are-these-continued-fractions-of-integrals-known) on MSE </sub>

**EDIT:** The period of the oscillations of $$f(s)=\dfrac1{1+\dfrac s{1+\dfrac{s^2/2!}{1+\dfrac{s^3/3!}{1+\cdots}}}}$$ appears to converge to $$\dfrac4e=1.4715\cdots$$ as @user42355 commented. Can this be proven?

Empirically, the argmins are

```
2.12897, 3.73409, 5.26940, 6.78494, 8.29119, 9.79168, 11.2881, 12.7814, 14.2723,
15.7613, 17.2487, 18.7348, 20.2198, 21.7038, 23.1870, 24.6695, 26.1513, 27.6326,
29.1133, 30.5937, 32.0736, 33.5531, 35.0323, 36.5111, 37.9897, 39.4680, 40.9461,
42.4239, 43.9015, 45.3790, 46.8562, 48.3332, 49.8101, 51.2869, 52.7635, 54.2399,
55.7162, 57.1924, 58.6685, 60.1445, 61.6203, 63.0961, 64.5717, 66.0473, 67.5228, ...
```
with consecutive differences (note insufficient precision beyond fourth decimal)
```
         1.60512, 1.53531, 1.51554, 1.50625, 1.50049, 1.49642, 1.49330, 1.49090, 
1.48900, 1.48740, 1.48610, 1.48500, 1.48400, 1.48320, 1.48250, 1.48180, 1.48130, 
1.48070, 1.48040, 1.47990, 1.47950, 1.47920, 1.47880, 1.47860, 1.47830, 1.47810, 
1.47780, 1.47760, 1.47750, 1.47720, 1.47700, 1.47690, 1.47680, 1.47660, 1.47640, 
1.47630, 1.47620, 1.47610, 1.47600, 1.47580, 1.47580, 1.47560, 1.47560, 1.47550, ...
```
-----

Define the continued fraction integral transform $$\{\mathcal If(t)\}(s)=\dfrac{f(s)}{1+\dfrac{\int_0^s f(x)\,dx}{1+\dfrac{\int_0^s\int_0^u f(x)\,dx\,du}{1+\cdots}}}$$ where $f\in C^\infty(\Bbb R)$. Has this been studied?

Elementary properties include $\{\mathcal I(0)\}(s)=0$, $\{\mathcal If\}(0)=f(0)$
 and $1+\{\mathcal If\}(s)=f'(s)/\{\mathcal If'\}(s)$ for all non-constant $f$ such that $f(0)=0$.

When $f(t)\equiv1$, the function $\{\mathcal I(1)\}(s)$, mentioned in [A319173](http://oeis.org/A319173) with no further information, has the following graph

[![enter image description here][1]][1]

Zooming out (orange with 140 terms; blue with 100 terms; magenta with 79 terms), we observe excellent convergence, until the plots branch off to either 0 or 1 depending on the parity of the number of terms.

[![enter image description here][2]][2]

How do we explain the regular oscillating behaviour over the negative reals?


  [1]: https://i.sstatic.net/bVKKE.png
  [2]: https://i.sstatic.net/yLA2f.png