Another technique, which is [discussed in this other Mathoverflow question][1] 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 $k$, $\mathrm{lcm}(1,2...k)|a_n$ if $a_n$ is sufficiently large, although I'm not aware of any natural example where $n!$ isn't the correct sequence to use here. Douglas Zare in that thread noted that this technique gives a nice proof that values of certain Bessel functions are irrational. 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