Another technique, which is discussed in this other Mathoverflow question 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. Douglas Zare in that thread noted 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?
JoshuaZ
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