Jonathan Sondow elegantly proves the irrationality of *e* in his aptly titled *A Geometric Proof that e Is Irrational and a New Measure of Its Irrationality* (The American Mathematical Monthly, Vol. 113, No. 7 (Aug. - Sep., 2006), pp. 637, <http://www.jstor.org/stable/27642006>).


In his argument, he constructs a sequence of nested intervals $I\_n$ for every $n \geq 1$, each of the form $[k/n!, (k+1)/n!]$, such that $\bigcap I\_n = \{e\}$, with $e$ lying strictly between the endpoints of each $I\_n.$ From this, we conclude that $e$ cannot be written as a fraction with denominator $n!$ for any $n \geq 1.$


**Fact:** Every rational number $p/q$ can be written as a fraction with a factorial in its denominator: $p/q = p(q-1)!/q!$. 


Thus, we conclude that $e$ is irrational.


The reason this proof technique works so well with $e$ is, of course, related to the Maclaurin series for the exponential function, $e^x.$


That any rational number can be written in lowest terms is employed in other irrationality proofs (e.g., the classic proof for that of $\sqrt{2}$) but I had not seen the above fact drawn upon before reading this particular paper.


**My question is:** are there other examples of real numbers (which are not related to $e$ in some trivial way) whose irrationality can be proved using the **Fact** above?



Irrationality proof technique: no factorial in the denominator
The same proof technique, for modified versions of the **Fact**, proves that some values of some hypergeometric functions are irrational. For example, the Bessel functions of the first kind have the following power series:


$$J\_n(x) = \sum\_{i=0}^\infty \frac{(-1)^i}{i! (i+n)!} \bigg(\frac x 2\bigg)^{2i+n} $$


For any choice of integers $m\ne 0$ and $n$, every rational number can be written as a quotient of integers so that the denominator is of the form $i!(i+n)!m^{2i+n}$. Since $J\_n(2/m)$ can't, it is irrational.


