Uniqueness summary and idea

[jaydfox]

If this is true, and an analytic iterate can be found by some other means, then can we perhaps derive an accuracy function?

These are two methods.
1. Practical mathematicians have developed a quite interesting theory how we can despite successfully use non-converging power series. However I am not familiar with theory, but the main point is that most non-converging series have a certain index k at which they are quite near the actual value of the function. If one can determine this k, one has an ultrafast approximation for the function.

I already alluded to this, but I'm apparently not very good at expressing myself. The index k depends on how close to the fixed point you are. As you get closer and closer to the fixed point, the index k increases without an upper bound. This can be seen by looking at the graph of the root tests that Andrew posted. Assuming the graph of the root tests continues linearly, or even exponentially, so long as it doesn't ever reach infinity, there is a radius for which the series initally converges, for all k. We need only prove that the root test never goes to infinity, and that doesn't seem like such a tall order, given how the coefficients are defined.

An exact solution can be found using a limit, with the radius going to 0, the index k going to infinity (as a function of the radius), and then using integer iteration counts (which we know are convergent) to analytically extend the radius back out to infinity. In a theoretic sense, as the limits are taken to their respective ends (0 and infinity), the solution is exact. From a computational/practical standpoint, you can find arbitrary precision with finite index k and a relative large radius (e.g., 0.001). If you can find the function that gives the correct index k for a given radius, you can explicitly compute what radius and what k are necessary to achieve a desired degree of precision.