How to calculate $x_{n}= \sqrt{n+ \sqrt{n- 1+ \sqrt{n- 2+ \sqrt{\cdots + \sqrt{2+ \sqrt{1}}}}}}$ without using surd signs

Question

How to calculate: $$x_{n}= \sqrt{n+ \sqrt{n- 1+ \sqrt{n- 2+ \sqrt{\cdots + \sqrt{2+ \sqrt{1}}}}}}$$ without using surd signs.

My attempt: I saw that $$x^{2}_{n}= n+ x_{n- 1}$$ Therefore $$x= 1+ \frac{n}{x}$$ So $$x= 1+ \frac{n}{1+ \frac{n- 1}{1+ \frac{n- 2}{1+ \frac{n- 3}{\ddots }}}}$$ But it seems like no much useful for small $n.$ I need to your helps to get it. Thanks a real lot !

Answer 1

First you can't apply the limit to

$$x_n^2=n+x_{n-1} \ \ (1)$$

to find the limit of $(x_n)$ because you have the divergent term $n$.

Note that

$$ \forall n,\ x_n >0 $$

So from $(1)$ you get

$$ \forall n , \ x_n > \sqrt{n}$$

So the sequence diverges.

Secondly closed form doesn't match well with the combination of plus sign and square-root so it is very unlikely to find an easier closed form.