Closed form for $\sqrt{x +\sqrt{x^2+\sqrt{x^3+\cdots}}}$

Asked Jun 30 '23 at 06:58

Active Jul 01 '23 at 05:37

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If this series $S_x$ converges for some $x$, can we find its closed form expression?

$$ S_x=\sqrt{x +\sqrt{x^2+\sqrt{x^3+\sqrt{x^4+\cdots}}}} $$

For $x=1$, it's easy and we have,

$$ S_1 = \sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\cdots}}}} \implies S_1^2=S_1+1 \implies S_1=\frac{1+\sqrt5}{2}$$

But I can't solve in an expression of $x$ for general case.

If the closed form is not known, can we prove that it does not exist ?

edited Jul 01 '23 at 05:37

asked Jun 30 '23 at 06:58

An_Elephant

5

Do you have a reason to believe it has a closed form? – Brian Moehring Jun 30 '23 at 07:15
2

See 1, 2, 3, 4 – Sil Jun 30 '23 at 07:29
@Ennar Thanks. I don't know why they didn't showed up as related duplicates while I was writing the question. – An_Elephant Jun 30 '23 at 07:46
@BrianMoehring No, I am not sure about that. See my first line : "can we find its closed form expression?" Thanks – An_Elephant Jun 30 '23 at 07:47
@Sil Thanks. I don't know why they didn't appeared while I writting the question. – An_Elephant Jun 30 '23 at 07:49
@An_Elephant, unfortunately, the search algorithm on MSE is far from ideal, many users use https://approach0.xyz/search/ to search up maths. – Ennar Jun 30 '23 at 08:00
@Ennar Thanks for link. Anyway, in any of the post , the conclusion is that we don't know it has closed form. Is there a proof that it don't have a closed form ? – An_Elephant Jun 30 '23 at 11:12
@An_Elephant, I can't tell you. If it's important to you, I'd ask a separate question (with all of these references) after I studied these links and gave a thought about it, and very importantly, explain in detail why the closed form is important to you and possibly others, but I'd guess it's not an easy question and you might not get a satisfactory answer. – Ennar Jun 30 '23 at 22:22
@Ennar Okay thanks. I'll edit the question accordingly. – An_Elephant Jul 01 '23 at 05:36

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