It is an interesting task to try finding the limit of nested square root expressions.
$$\lim_{n \to \infty}\left( 1 + \sqrt{2 + \sqrt{3+ ... + \sqrt {n + \sqrt{n+1}}}}\right)$$
How to solve this one?
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vladon
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Relevant: This and this. – Ben Jan 28 '14 at 12:19
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Convergence of this nested radical expression can be seen by Herschfeld's convergence test (see Herschfeld, On Infinite Radicals. Amer. Math. Monthly 42, 419-429, 1935.):
Theorem: For $0<p<1$ and $a_n\ge 0$, the limit $$\lim_{n\rightarrow\infty} a_1+(a_2+(\cdots+(a_n)^p)^p)^p$$ exists if and only if the sequence $(a_n^{p^n})_n$ is bounded.
That reduces checking convergence to seeing that $a_n^{p^n}=n^{2^{-n}}$ is bounded, which is clear since
$$n^{2^{-n}}=e^{2^{-n}\log n}\longrightarrow 1$$
as $n\rightarrow\infty$.
However, no closed form is known to express the limit.
J.R.
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This is probably a long shot, but since you mentioned Herschfeld, I have to ask: is there any deeper meaning to his observation that continued fractions are nested radicals of order $-1$ ? Some implications, perhaps ? Thank you. – Lucian Jan 28 '14 at 13:01
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@Lucian No, as far as I remember he just remarks that and nobody really knows what that is good for. At least I'm not aware of any implications. – J.R. Jan 28 '14 at 13:39
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This is the square of the Nested Radical Constant, which converges, but is not known to possess a closed form. See also Somos's Quadratic Recurrence Constant.
Lucian
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