Another technique, which is discussed in the Mathoverflow question https://mathoverflow.net/questions/103129/irrationality-proof-technique-no-factorial-in-the-denominator is to show that $n!$ cannot be the denominator for any $n$. In principal you could use this technique not for $n!$ but for any sequence of denominators $a_n$ where for any $k$, $\operatorname{lcm}(1,2,\dotsc,k)\mid a_n$ if $a_n$ is sufficiently large. Douglas Zare in that thread [noted](https://mathoverflow.net/a/103161) that this technique gives a nice proof that values of certain Bessel functions are irrational with a specific chosen set that is not $n!$ but arises from the series. It seems like this works mainly for fast-converging series, so this may be somehow that technique in disguise? 

  [1]: https://mathoverflow.net/questions/103129/irrationality-proof-technique-no-factorial-in-the-denominator?rq=1