jaydfox Wrote:I already have code in place to remove the "correct" singularity, and I could just modify it to instead introduce a false set of coefficients for k>n, and see what happens. As I think more about what a partial series (only terms k>n) of a singularity would look like as a function, i.e., an increasingly insignificant singularity, I'm actually going to guess that the solution will still converge on the correct one, even if only very, very slowly. (And convergence with v_k=0, k>n, is already very slow.)
Hmm, just thinking about practicalities, however, the root test of any false function must be less than 1, as is trivially obvious from the first row, but also from the fact that we're solving across an interval (0,1), so we need a root test at most 1 to reach the end of the interval. But I can try introducing false singularities just outside this radius and see what happens. I'll start with Henryk's suggestion, then try a singularity outside the correct radius of convergence, then one at the radius but at the wrong location, then one inside the correct radius (if there's even a need to go that far).
Update: I'm going to further venture to hypothesize that any series we use for k>n must have a radius of convergence greater than that of the true singularity. If it has the same radius of convergence, then that power series must converge on the location(s) of the true singularity(ies), perhaps even the form of the singularity as well (e.g., logarithmic versus simple pole).
I came to this idea as I was thinking about cyclic alterations of the correct series (e.g., \( G(z) = F(z) + \frac{\sin\left(2\pi F(z)\right)}{200\pi} \)). They should satisfy the Abel equation, and hence I wouldn't expect the series as n grows to change from that series to the correct one, because as n grows, we can just plug in the modified series and get a solution, and the finite matrices should have unique solutions.
On the other hand, for any singularity outside the radius of convergence, as n grows the terms would eventually become too small relative to the terms of the correct sequence, and hence they would become increasingly irrelevant. Therefore, I would still expect the series to converge on the correct sequence in those cases.
~ Jay Daniel Fox