The infinite series $1-1+1-1+\cdots$ diverges because the sequence of partial sums, $1,0,1,0,\ldots$ has no limit. However, it is well know that we can get around this problem in a number of ways; the series is summable using alternate methods, such as the Cesaro sum $$c_{n}=\frac{1}{n}\sum_{i=1}^{n}s_{n}=\frac{1}{n}\sum_{i=1}^{n}\sum_{j=1}^{i}a_{n}.$$ For the series mentioned above, the Cesaro summation gives $\frac{1}{2}$, which is also the analytic extension $\lim_{x\rightarrow-1}\frac{1}{1-x}$. More elaborate summation methods can of course be used to regularize even trickier series.

However, I am having some trouble dealing with non-convergent sequences that do not arise as series sums, but which do still seem to have some "structure" that might be used to regularize them. The real-valued sequence $1,0,1,0,\ldots$ arose as the partial sums in the example in the previous paragraph, but that same divergent sequence arises if you are trying to calculate a limit of
$$\sqrt{1-\sqrt{1-\sqrt{1-\sqrt{1-\cdots}}}}.$$
Becuase $1,0,1,0,\ldots$ does not converge, this limit of nested radicals does not exist—*but if it did exist*, it seems that it ought to converge to $\phi^{-1}$, the inverse of the Golden Mean. $\phi$ itself is, of course, a similar limit
$$\phi=\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\cdots}}}},$$
since this is clearly convergent and satisfies $\phi^{2}=1+\phi$. The purported nested radical for $\phi^{-1}$ would obviously satisfy $\left(\phi^{-1}\right)^{2}=1-\phi^{-1}$, if only it converged.

So what I am asking is whether there is a technique for regularizing $\sqrt{1-\sqrt{1-\cdots}}$ to get $\phi^{-1}$ that can be expressed as some kind of (non-arithmetic) average over the sequence $1,0,1,0,\ldots$. I think it is clear that one can concoct an *ad hoc* averaging procedure to produce an regularized version of that sequence that will converge to any value in $(0,1)$, but I am looking for one that would somehow encode the information that the sequence arose from a nested radical expression—so that it could, I hope, be extended to other similarly divergent nested radicals.

Divergent Seriesis the classic compendium of this material. $\endgroup$