Definition of convergence of a nested radical $\sqrt{a_1 + \sqrt{a_2 + \sqrt{a_3 + \sqrt{a_4+\cdots}}}}$?

In my answer to the recent question Nested Square Roots, @GEdgar correctly raised the issue that the proof is incomplete unless I show that the intermediate expressions do converge to a (finite) limit. One such quantity was the nested radical $$ \sqrt{1 + \sqrt{1+\sqrt{1 + \sqrt{1 + \cdots}}}} \tag{1} $$

To assign a value $Y$ to such an expression, I proposed the following definition. Define the sequence $\{ y_n \}$ by: $$ y_1 = \sqrt{1}, y_{n+1} = \sqrt{1+y_n}. $$ Then we say that this expression evaluates to $Y$ if the sequence $y_n$ converges to $Y$.

For the expression (1), I could show that the $y_n$ converges to $\phi = (\sqrt{5}+1)/2$. (To give more details, I showed, by induction, that $y_n$ increases monotonically and is bounded by $\phi$, so that it has a limit $Y < \infty$. Furthermore, this limit must satisfy $Y = \sqrt{1+Y}$.) Hence we could safely say (1) evaluates to $\phi$, and all seems to be good.

My trouble. Let us now test my proposed idea with a more general expression of the form $$\sqrt{a_1 + \sqrt{a_2 + \sqrt{a_3 + \sqrt{a_4+\cdots}}}} \tag{2}$$ (Note that the linked question involves one such expression, with $a_n = 5^{2^n}$.) How do we decide if this expression converges? Mimicking the above definition, we can write: $$ y_1 = \sqrt{a_1}, y_{n+1} = \sqrt{a_{n+1}+y_n}. $$ However, unrolling this definition, one get the sequence $$ \sqrt{a_1}, \sqrt{a_{2}+ \sqrt{a_1}}, \sqrt{a_3 + \sqrt{a_2 + \sqrt{a_1}}}, \sqrt{a_4+\sqrt{a_3 + \sqrt{a_2 + \sqrt{a_1}}}}, \ldots $$ but this seems little to do with the expression (2) that we started with.

I could not come up with any satisfactory ways to resolve the issue. So, my question is:

How do I rigorously define when an expression of the form (2) converges, and also assign a value to it when it does converge?

Thanks.

Solution 1:

Vijayaraghavan proved that a sufficient criterion for the convergence of the following sequence $\ \sqrt{a_1 + \sqrt{a_2 +\:\cdots\: +\sqrt{a_n}}}\ \ $ is that $\displaystyle\ \ {\overline {\lim_{n\to\infty}}}\ \frac{\log\:{a_n}}{2^n}\: < \:\infty\:.\: $

For references see see this 1935 Monthly article, Herschfeld: On infinite radicals, and Raoa and Berghe: On Ramanujan's nested roots expansion 1, 2005 and see this prior answer.

Solution 2:

I would understand it by analogy with continued fractions and look for a limit of $\sqrt{a_1}$, $\sqrt{a_1+\sqrt{a_2}}$, $\sqrt{a_1+\sqrt{a_2+\sqrt{a_3}}}$, ..., $\sqrt{a_1+\sqrt{a_2 \cdots + \sqrt{a_n}}}$, ...

Each of these is not simply derivable from the previous one, but neither are continued fraction approximants.