I'm hoping to investigate the effects of divergent summation methods on series which cannot be analytically continued due to a dense set of singularities. At least *a priori*, it doesn't seem that a natural boundary should necessarily mean there aren't analytic functions that continue the original function. For example, we can easily generate somewhat contrived examples such as $\sum_{n=0}^\infty \frac{1}{2^{n-1}(n+1)^2}\sum_{k=0}^{2^{n-1}} (z-e^{\frac{2k+1}{2^n} \tau i})^{-1}$ This series absolutely converges whenever z is not exactly equal to a dyadic rational. On the other hand, it has a natural boundary at the unit circle because the dyadic rationals are dense on the unit circle.

So because the terms in this series eventually go to zero, the partial sums provide arbitrarily good approximations of the function when not around the unit circle, and so we have a natural way to "continue" this function to an analytic function using the partial sums.

We can easily craft many of these examples by enumerating the rationals--or some other dense set-- in steps and causing the later steps to eventually go to 0. Each of these functions will similarly have a natural boundary, yet can be approximated by analytic functions.

I'm interested in finding the validity of using divergent series methods to find continuations of functions with natural boundaries. For instance, one function I am looking at is $$\sum_{n=0}^{\infty}\frac{(-1)^{n}x^{\frac{n(n+1)}{2}}}{\prod_{m=1}^{n}(1+x^{m})}$$ which is from one of Ramanujan's notebooks. It is special in that its terms grow quite slowly, so it's easier to handle with divergent series methods.

In particular, applying different types of smooth cutoff functions to this function leads to the same thing. Furthermore, the smooth cutoff method agrees with my approximation method. Graphed together, these methods look like this:

The green line is the smooth cutoff function (using $\frac{1}{1.005^{n^2}}$), the orange line is the approximate method and blue is approximating Borel method (which ends up essentially like applying a smooth cutoff function). Each of these functions agrees with the starting function when the starting function converges.

Are there other ways to regularize this series that lead to this same result? Are there other methods that disagree? Also, because we don't have the identity theorem when we allow natural boundaries, is there a natural replacement of this theorem, possibly using properties of divergent series? My guess is that something like the Tauberian theorems could provide a good start for proving some sort of uniqueness, though it's not ideal because it requires somewhat strong restrictions on the properties of the terms in the summation.

I'd appreciate any insight into this problem, or insight into similar problems with faster divergences (for instance, I'm also interested in $\sum_{n=0}^\infty (-1)^n x^{n^2}$).

Finally, are there are any other proposed methods of continuing functions with natural boundaries that I can check the divergent series methods against? It's quite useful for me to be able to check my own results against the results of others to find where the methods agree and disagree, and hopefully why.