Calculate Limit 0f nested square roots

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?

• Relevant: This and this. – Ben Jan 28 '14 at 12:19

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.

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.

• 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
• @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