Solving a radical to find the value

Question

How to find the value of this nested radical expression ?

$$\sqrt{7+\sqrt{7-\sqrt{7+\sqrt{7-\cdots\infty}}}}$$

Answer 1

This seems like the only reasonable interpretation to me: \begin{align*} x_{0} & =0\\ x_{n} & =\sqrt{7+\sqrt{7-x_{n-1}}} & \text{for }n\geq1. \end{align*} If this iteration has a fixed point $x$, it must satisfy $$ x=\sqrt{7+\sqrt{7-x}}. $$ You can show that $x=3$, but I am lazy.

Answer 2

Let $x=\sqrt{7+\sqrt{7-\sqrt{7+\sqrt{7-..\infty}}}} \ \ \ (1)$

Put to the square:

$$x^2=7+\sqrt{7-\sqrt{7+\sqrt{7-..\infty}}}$$

Isolate the radical in the RHS:

$$x^2-7=\sqrt{7-\sqrt{7+\sqrt{7-..\infty}}}$$

Remark: note that $x^2-7$ has to be positive.

Put to the square again:

$$(x^2-7)^2=7-\sqrt{7+\sqrt{7-..\infty}}$$

But the expression under the radical sign is exactly that in (1), i.e., $x$.

Thus $x$ verifies equation:

$$(x^2-7)^2=7-x$$

which is equivalent to $(x-3)(x+2)(x^2+x-7)=0$

with roots $x=-2, x=3, x=\dfrac{-1+\sqrt{29}}{2}=2.19258, x=\dfrac{-1-\sqrt{29}}{2}=-3.19258$

This equation has only one acceptable positive root (i.e. such that the remark above is fulfilled) :

$$x=3$$

Edit: I have assumed that the convergence issue is adressed elsewhere.

Nevertheless, here is a way to do it.

Define a vector sequence $(a_n,b_n)^T$ by: $(a_1,b_1)=(1,1)$ and

$$\begin{cases}a_{n+1}&=&\sqrt{7+b_n}\\b_{n+1}&=&\sqrt{7-a_n}\end{cases}$$

with fixed point $(a,b)=(3,2)$ and $|det(Jac(F))|<1/24$ at the fixed point, explaining a very rapid convergence.

Jac(F) = Jacobian matrix of associated transformation $F$:

$F:(x,y) \rightarrow (u,v)$ with $u=\sqrt{7+y}, v=\sqrt{7-x}$

Moreover, there is a rather large domain of convergence, that is $D=[1,6]\times[1,6]$ which is such that $F(D) \subset D$ (see remark below) where :

$|det(Jac(F))|=\dfrac{1}{4 \sqrt{(7+y)(7-x)}}\leq\dfrac{1}{8 \sqrt{2}}$

Remark : inclusion $F(D) \subset D$ can be obtained at once by considering the variations of monotonous functions defined by $f(x)=\sqrt{7+x}$ and $g(x)=\sqrt{7-x}$ on interval $[1,6]$.

Here is a picture displaying the rapid convergence to the limit point $(3,2)$ for different initial points $(a_1,b_1) \in D$.