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?

Solution 1:

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.

Solution 2:

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.