Interesting & beautiful Series/Products/Nested Radicals/Continued Fractions

Question

I've encountered various stunning series, products, nested radicals and continued fractions, such as the ones listed below: $$\frac{\pi}{4}=1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+...$$ $$\frac{\pi^2}{6}=\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...$$ $$e=1+\frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!}+...$$ $$\frac{2}{\pi}=\frac{\sqrt2}{2}\times\frac{\sqrt{2+\sqrt2}}{2}\times\frac{\sqrt{2+\sqrt{2+\sqrt2}}}{2}\times...$$ $$3=\sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{1+...}}}}$$ I've been interested in finding some other stunning series, products, nested radicals or continued fractions; if you know any, would you mind posting them below?

Thank you.

Please note, I do not want series or products etc that are polynomials, eg Taylor/Maclaurin series (beautiful as they may be!). I would only like expressions with constant terms.

I noticed that someone voted to close the question. If you have any similar feelings, can you please comment why? — A-Level Student, Oct 16 '20 at 14:37
@A-levelStudent I can agree that the formulas involving $\pi$ are beautiful, but the ones relating to $e$ and $\phi$ are, in my opinion, just the definitions of $e$ and $\phi$... (I did not vote to close, however) — TheSilverDoe, Oct 16 '20 at 15:33
@TheSilverDoe you have a fair point there, I will change them. Incidentally, is there any other infinite series that has been found to calculate $e$? — A-Level Student, Oct 16 '20 at 15:40
Do not care about one close-vote (not mine). This is a somewhat subjective question , this is probably the reason for the close-vote. But you have a $+2$-score, so currently you need not worry. — Peter, Oct 16 '20 at 15:53
Since "stunning" is not a mathematical notion, you'd better ask this elsewhere. No, seriously, this is just attracting personal opinions. — , Oct 16 '20 at 18:03
@ProfessorVector I don't think that's fair. I think all mature mathematicians can agree on whether an equality is stunning or not. — A-Level Student, Oct 17 '20 at 20:33
@Gerben everyone is entitled to their own opinion :) I am afraid I heartily disagree with you. Formulae like $e^{i\pi}+1=0$ are easily proved but are still constantly praised and adored by the mathematical community. See here: https://en.wikipedia.org/wiki/Euler%27s_identity — A-Level Student, Oct 19 '20 at 19:25
Nah do your thing and enjoy, be happy and cheerful :) it's just not my thing. — Gerben, Oct 19 '20 at 21:34

Interesting & beautiful Series/Products/Nested Radicals/Continued Fractions

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