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Another technique, which is discussed in the Mathoverflow question 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 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?

Another technique, which is discussed in the Mathoverflow question 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 $n$ 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?

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?

Another technique, which is discussed in the Mathoverflow question 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 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?

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, 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?

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?