The dimensionless fine structure constant $\alpha \approx \frac1{137}$ has intrigued physicists for over a century. Whilst not currently a majority view, there is a school of thought that considers this constant may have a mathematical origin, just like $\pi$ or $e$. Many researchers have indeed proposed formulas for $\alpha$ in the past (including on this forum), all of which have progressively fallen outside of experimental bounds as the fine structure constant was measured with ever more precision.
In 1971, for example, Physics Today reviewed the work of Armand Wyler, who proposed $\alpha = (2^{19}3^{-7}5\pi^{11}5)^{1/4}$ yielding approximately 1/137.03608, which was close to the experimental value at the time of 1/137.03611 (21). The article elicited many replies, among which half a dozen of alternative combinations of powers of 2, 3, 5 and $\pi$ which all approximated $\alpha$ with similar precision, or better. One reader went on to show mathematically that Wyler's precision was not unusual, given the high integer powers used in the formula.
In 2004, Hans de Vries proposed a rather different type of formula for $\alpha$, which reads: $$ \alpha \sqrt{e^{\pi^2}}=\left(1+\frac{\alpha}{(2\pi)^0}\left(1+\frac{\alpha}{(2\pi)^1}\left(1+\frac{\alpha}{(2\pi)^2}\left(1+...\right)\right)\right)\right)^2 $$
There's an intuitive appeal to this formula, as it uses only the expected mathematical constants and a minimum amount of free integers. Its value, 0.0072973525686..., is within experimental accuracy of the current recommended value at NIST: 0.0072973525693 (11).
My question is: what can we say about the likelihood that this is a numerical coincidence? For example, how straightforward would it be to construct another 'simple' infinite series that converges onto the same experimental value range?
The fine structure constant is measured more and more precisely by different methods every so often. The value mentioned here is the current recommended value from NIST at the time of writing. The latest NIST value may be accessed online.
See also this answer to a related question, where the user states
If you can find a simple mathematical recipe to arrive at this constant, this would be noteworthy.
The Fine-Structure Constant: From Eddington's Time to Our Own, Jacob. D. Bekenstein in The Prism of Science, 1986.
A mathematician's version of the fine structure constant, Gloria Lubkin in Physics Today, August 1971
A new pastime — calculating alpha to one part in a million, Letters to the editor in Physics Today, November 1971
Based on an back-of-the-envelope information-theoretical calculation, I don't think there's anything particularly surprising here. The measured value and the solution to that expression for $\alpha$ share $9$ digits. That implies about $9 \log_2 10 \approx 30$ bits of information.
Compare that with the parse tree of the expression (I used your summation form). The way I construct it—maybe I'll draw it out and paste it in, but I doubt it, haha—it has $29$ nodes. These nodes contain things like
Even optimized for their frequency in this particular expression, this tree requires at least $62$ bits to encode. The fact that it produces $30$ bits of accuracy is therefore unsurprising. That doesn't mean that it isn't "correct" (whatever that means), but its correctness should not be presumed on probabilistic grounds.