Nested Radicals which is approximately 1.75

Question

We know that from using Ramanujan formula, we have :
$$3 = \sqrt{1+2\sqrt{1+\sqrt{3+\cdots}}}$$ Now, suppose I define the following sequence : $$ x_1 = 1\\ x_2 = \sqrt{1+\sqrt{2}}\\ \vdots\\ x_n = \sqrt{1+\sqrt{2+\sqrt{3+\cdots}}} $$ I have checked numerically the the value for $x_n$ is approximately $1.7$ but it does not equate to $\sqrt3$.

Can anyone help me to solve the exact value of this sequence?

Answer 1

The value to which the sequence converges is called the Nested radical constant, no closed-form expression is known for this constant.

Answer 2

Here is perhaps an approach. Let $$ A_k = \sqrt{k + \sqrt{k+1 + \sqrt{\ldots}}} $$ then assuming $(A_k)$ exists, $$ A_{k+1} = A_k^2-k $$ and $A_1$ will give you what you are seeking.