Last evening, after reading a couple of questions about nested radicals, I started to wonder about problems involving what I will term "alternating nested radicals;" below is an example, which I found here.
Prove the convergence of, and evaluate, $\sqrt{7 -\sqrt{7 + \sqrt{7...}}}$
It turns out that this nested radical converges to $2$, and this is not especially hard to argue on an ad-hoc basis. However, I became interested in the general solution of the problem of evaluating $\sqrt{q -\sqrt{q + \sqrt{q...}}} \text{ }\text{ }$ for an arbitrary positive real $q$. Despite some searching, I was not able to find a paper which stated a theorem on this, so I resorted to working it out for myself.
I have developed an argument which I believe gives the correct result for all $q > 1.$ I have checked its predictions against several alternating nested radicals, and they agree with computation.
Theorem: $\sqrt{q -\sqrt{q + \sqrt{q...}}} = \frac{\sqrt{1 + 4(q-1)}-1}{2}$
Argument: We make two significant assumptions:
- that $\sqrt{q -\sqrt{q + \sqrt{q...}}} \text{ }\text{ }$ converges; and,
- inspired by the fact that $\sqrt{q +\sqrt{q + \sqrt{q...}}} \text{ }\text{ } = x$ satisfies $x^2 - x - q = 0$, we assume that $\sqrt{q -\sqrt{q + \sqrt{q...}}} \text{ }\text{ } = x$ satisfies $x^2 + x - a$ for some $a$.
By the self-similarity of the alternating nested radical, we find
\begin{align} \sqrt{q - \sqrt{q + x}} &= x \\ q - \sqrt{q + x} &= x^2 \\ &= a-x \\ \therefore q + x - \sqrt{q + x} &= a \end{align}
Solving this as a quadratic in $\sqrt{q+x}$ yields $$\sqrt{q+x} = \frac{\sqrt{1 + 4a} + 1}{2}.$$ Again using our second assumption, we have $$x = \frac{\sqrt{1 + 4a} - 1}{2}.$$ But now
$$ \sqrt{q+x} - x = \frac{\sqrt{1 + 4a} + 1}{2} - \frac{\sqrt{1 + 4a} - 1}{2} = 1$$
and so we get
$$ q+x = (1+x)^2$$
and thus
$$ x^2 + x - (q-1) = 0.$$
Solving this quadratic for $x$ then gives the theorem.
I am pretty sure that my second assumption above would be impossible to defend; so, basically, I have the following questions:
- How can we rigorously prove that the alternating nested radical converges?
- How can we show, for a general $q$, that it converges to the value given in the theorem?
And one other thing: any general references to ubiquitous convergence theorems or techniques would be greatly appreciated!