My current research has me working with series of the form $\sum_{n=0}^\infty f_n z^n$, $z\in\mathbb{C}$ that usually have zero convergence radius (they diverge for all values of $z$). Due to unrelated reasons, I need to assume that each of these series is asymptotic to some function $f(z)$ that is well-defined near $z=0$ (not necesarily at $z=0$): $f(z)\sim \sum_{n=0}^\infty f_n z^n$.
As it is stated here (under the uniqueness section), a series can be asymptotic to many functions, and the best possible estimate is the finite function $\sum_{n=0}^N f_n z^n$, for $N>0$ such that the truncation error $\mathcal{E}(z,N)\equiv f(z)-\sum_{n=0}^N f_n z^n$ is minimum. Obviusly, if one does not know $f(z)$, $\mathcal{E}(z,N)$ cannot be computed and minimized.
It is a customary practice in my field (and it is also mentioned in the link above) to believe that the procedure of Borel summation (for which $f(z)$ does not need to be known) returns the best possible estimate of $f(z)$ that can be constructed with the series $\sum_{n=0}^\infty f_n z^n$. However I have found no formal proof of this, and several questions arise:
- Does the Borel sum return the same estimate one would obtain if $f(z)$ was known and $\mathcal{E}(z,N)$ could be minimized? Or is it "the best possible estimate given $f(z)$ is not known"?
- In any of the two above cases, is there any proof?
- The link indicates Watson's theorem and Carelman's theorem do show Borel summation produces the best estimate, but I fail to see how.
I am only interested in series whose coefficients $f_n$ grow as $a^n n^b n!$ for $a,b$ constants, so a proof that Borel summation is returns the best estimate for that class of divergent series is enough.
Thanks in advance!