The following is a problem inspired from quantitative program verification. It is more vague than ``Prove or disprove whether XYZ holds'', but in turn has real applications in probabilistic verification. I posed this problem several times to several different machine learning experts and none of them were able (or interested) to help me with this problem. I now hope that the automata (learning) experts can help me out.

We would like to learn / guess the limit of the sequence
\[ 1, \quad \frac{3}{2}, \quad \frac{7}{4}, \quad \frac{15}{8}, \quad {\dots} \quad {}\longrightarrow{} \quad 2
\] In order to approach this problem more systematically, we do have – in
practice – access to a more symbolic representation of this sequence, namely
\[ 1, \quad 1 + q, \quad 1 + q + q^2, \quad 1 + q + q^2 + q^3, \quad {\dots} \quad {}\longrightarrow{} \quad \frac{1}{1 - q}~.\]
The limit \(\frac{1}{1-q}\) has Taylor expansion
\[ 1 + q + q^2 + q^3 + q^4 + q^5 + \cdots
\] So learning the limit of this sequence appears like wanting to learn the
``regular language''~\(q^*\) but only from positive examples: First we get \(1 ({=}
\varepsilon)\), then \(q\), then \(q^2\), and so on. So the problem is: **How can we
learn / guess regular languages from positive examples (reasonably well)?**

Going one step further, we would also like to learn the following sequence (which did not occur to me in practice, but I made up): \[ q, \quad q + 2q^2, \quad q + 2q^2 + 3q^3, \quad q + 2q^2 + 3q^3 + 4q^4, \quad {\dots} \quad {}\longrightarrow{} \quad \frac{1}{(1 - q)^2}\] The limit \(\frac{1}{(1-q)^2}\) has Taylor expansion \(q + 2q^2 + 3q^3 + 4q^4 + 5q^5 + \cdots\)

So learning the limit of this sequence appears like wanting to learn a
*weighted* ``regular language'' (are there regular expressions for weighted
languages?) but only from positive examples. So: **How can we learn /
guess weighted regular languages from positive examples (reasonably well)?**

Another example – this one again does actually occur in practice – is learning
the limit of the (more complicated) sequence
\[ 1, \quad 2, \quad 2 + 2q - q^2, \quad 2 + 2q + 2q^2 - 2q^3, \quad {\dots} \quad {}\longrightarrow{} \quad
\textnormal{???}
\]
Notice that, for \(q^2\), the weight changed from \({-}1\) to \(2\) from iteration 3
to 4. The same can be observed for the \(q^0\) weight from iteration 1 to 2. I
hence suspect that this is not ``learning from *positive* examples''
anymore, but something slightly more general. In particular, I do believe that
the weights for each \(q^n\) stabilize at some iteration and never again change.
So our problem is: **From what kind of examples are we even trying to
learn here? Can such learning be done (reasonably well)?**