Given the following infinite series with rational $x < 1$ && integer $n \geqslant 2$:
$$1-\frac{x}{n}+\frac{(1+n)x^2}{2n^2}-\frac{(1+n)(1+2n)x^3}{6n^3}+\frac{(1+n)(1+2n)(1+3n)x^4}{24n^4}-\ldots$$
Is it possible to prove certain values of $x, n$ will converge to a irrational result? I can generate an infinite generalized continued fraction with Euler's formula, but after transforming $a_i, b_i$ to positive integers $a_i>b_i$ which does not indicate whether the result is irrational or rational.
It seems that a regular continued fraction makes the determination trivial: Finite fractions are rational and infinite convergent fractions are irrational. Is there an algorithm/heuristic for converting to a regular continued fraction, or is that a manual process?