I hope this is not a duplicate, this is quite a general question but I couldn't find the answer in any of the other posts.
Suppose one has a power series $\sum_{n=0}^{\infty}c_n x^n$ whose coefficients $c_n$ can only be numerically computed (for example, in my particular case, the $c_n$ are obtained by inverting a matrix). The general question here would be how to implement the usual convergence tests (ratio test and root test specifically) to extract whether the series has a finite or inifite convergence radius in $x$.
To be more specific on the problem of having access to only the numeric values of $c_n$, I'll show my attempts with the root and ratio tests. I compute the first 150 $c_n$ coefficients and plot $\vert c_n\vert^{1/n}$ as a function of $n$:
To me this picture does not make it clear whether or not the sequence approaches a finite value. With a little effort, I compute more coefficients, this time I go to 450. This is the result
Which looks as inconclusive as the previous picture, as I cannot tell if the sequence approaches a value at around $2\pi$ or grows to infinity.
Interestingly, the ratio test looks like this:
Where the sequence that I plotted is $\vert c_{n+1}/c_n\vert$. It seems that the "sub-sequence" given by odd n+1 and even n goes to zero while that given by even n+1 and odd n goes to infinity. I am unsure if the presence of the seemingly divergent contribution is a reason strong enough to claim the power series has zero radius of convergence.
In sum, I am having trouble extracting strong conclusions about the convergence radius of the power series from this kind of analysis, and I was wondering if there are methods that can be applied other than plotting a few terms and trying to guess the general behavior.
Edit: from the comments I see there is in principle no way of determining the convergence radius by looking at a finite number of terms. This is already an answer, but I'll leave here the specific problem that motivated the question.
Let us consider the matrix $S$, whose coefficients $s_{n,k}$ are
\begin{equation} \begin{aligned} s_{n,k}&=0,\quad k<n,\\ s_{n,k}&=\frac{-i(-1)^{k+n} \Gamma (k)}{2^k(i\pi)^{n}\Gamma^2(n) \Gamma(k-n+2)}\frac{f(n,k)}{g(n)},\quad k\geq n, \end{aligned} \end{equation}
where
\begin{equation} \begin{aligned} f(n,k)&=(2i\pi)^n \Gamma(k+1,-2 i \pi ,2 i \pi )\\& +(2i\pi)^{k+1}[(-1)^{k+n} (\Gamma (n,-2 i \pi )-\Gamma (n))+\Gamma (n,2 i \pi )-\Gamma(n)],\\ g(n)&=2\pi(\tilde{\Gamma}(n,-2 i \pi )+\tilde{\Gamma}(n,2 i \pi)-2)+i n \tilde{\Gamma}(n+1,2 i \pi,-2i\pi). \end{aligned} \end{equation}
In these expressions $i$ is the imaginary unit, $\Gamma(z,a,b)$ is the incomplete gamma function and $\tilde{\Gamma}(z,a,b)=\Gamma(z,a,b)/\Gamma(z)$.
The $S$ matrix is upper triangular, and all its diagonal elements are $1$. Let us now consider the inverse matrix $T\equiv S^{-1}$. Then the $c_n$ coefficients of the power series above are the elements of the first row of $T$.
In these conditions I have not been able to find a form for the entries $t_{n,k}$ of the $T$ matrix, and thus I've been forced to resort to generate $S$ numerically and invert it.
