# More on irrationality

Recall some of the irrationality criteria that we discussed in the last post. We showed that

Proposition 1. Let $a_1,a_2,\dots$ be a sequence of non-zero integers such that

$(\dagger)\qquad\displaystyle S=\frac{1}{a_1}+\frac{1}{a_1a_2}+\frac{1}{a_1a_2a_3}+\dots$ exists, and

$(\ddagger)\qquad\displaystyle \frac{1}{a_{n+1}}+\frac{1}{a_{n+1}a_{n+2}}+\frac{1}{a_{n+1}a_{n+2}a_{n+3}}+\dots\to 0$ as $n\to\infty$.

Then $S$ is irrational.

In particular,

Proposition 2. If $a_1,a_2,\dots$ is a sequence of non-zero integers such that $|a_1|\le |a_2|\le\dots$ and $\displaystyle\lim_{n\to\infty}|a_n|=\infty$, then

$\displaystyle S=\frac{1}{a_1}+\frac{1}{a_1a_2}+\frac{1}{a_1a_2a_3}+\dots$

exists and is irrational.

Having proven these, it is natural to ask whether every irrational number can have such a representation. Interestingly, work has already been done on this. The Engel expansion of a positive real number $x$ is a unique expansion of the form

$\displaystyle x=\frac{1}{a_1}+\frac{1}{a_1a_2}+\frac{1}{a_1a_2a_3}+\dots$

where $\{a_n\}$ is a non-decreasing sequence of positive integers. Every positive rational number has a finite Engel expansion, and $x$ is irrational if an only if this expansion is infinite.

In this post we shall slightly improve our previous results.

Proposition 3. Let $a_1,a_2,\dots$ and $b_1,b_2,\dots$ be sequences of non-zero integers such that

$(\dagger')\qquad\displaystyle S=\frac{b_1}{a_1}+\frac{b_2}{a_1a_2}+\frac{b_3}{a_1a_2a_3}+\dots$ exists, and

$(\ddagger')\qquad\displaystyle \frac{b_{n+1}}{a_{n+1}}+\frac{b_{n+2}}{a_{n+1}a_{n+2}}+\frac{b_{n+3}}{a_{n+1}a_{n+2}a_{n+3}}+\dots\to 0$ as $n\to\infty$.

Then $S$ is irrational.

Proof. The same argument in the proof of Proposition 1 applies. $\square$

Proposition 4. If $a_1,a_2,\dots$ and $b_1,b_2,\dots$ are sequences of non-zero integers such that $|a_1|\le |a_2|\le\dots$ and $\displaystyle\lim_{n\to\infty}|b_n/a_n|=0$ then

$\displaystyle S=\frac{b_1}{a_1}+\frac{b_2}{a_1a_2}+\frac{b_3}{a_1a_2a_3}+\dots$

exists and is irrational.

Proof. It suffices to show that $(\dagger')$ and $(\ddagger')$ above hold.

Convergence follows easily using the ratio test. So we show that $(\ddagger')$ holds.

We have, for sufficiently large $n$,

$\displaystyle\left|\frac{b_{n+1}}{a_{n+1}}+\frac{b_{n+2}}{a_{n+1}a_{n+2}}+\frac{b_{n+3}}{a_{n+1}a_{n+2}a_{n+3}}+\dots\right|$

$\displaystyle\le\left|\frac{b_{n+1}}{a_{n+1}}\right|+\left|\frac{b_{n+2}}{a_{n+2}}\right|\frac{1}{|a_{n+1}|}+\left|\frac{b_{n+3}}{a_{n+3}}\right|\frac{1}{|a_{n+1}a_{n+2}|}+\dots$

$\displaystyle\le\left|\frac{b_{n+1}}{a_{n+1}}\right|+\frac{1}{|a_{n+1}|}+\frac{1}{|a_{n+1}|^2}+\frac{1}{|a_{n+1}|^3}+\dots$

$\displaystyle =\left|\frac{b_{n+1}}{a_{n+1}}\right|+\frac{1}{|a_{n+1}|-1}$

$\displaystyle\le\left|\frac{b_{n+1}}{a_{n+1}}\right|+\frac{2}{|a_{n+1}|}$

$\displaystyle\le 3\left|\frac{b_{n+1}}{a_{n+1}}\right|\to 0$ as $n\to\infty$.

So $(\ddagger')$ holds, as desired. $\square$

As an immediate corollary we get:

Corollary 1. Let $f:\mathbb R\to\mathbb R$ have an infinite Taylor expansion about $x=0$ that converges for $|x|\le 1$. If $f^{(n)}(0)\in\mathbb Z\,\forall n\ge 0$ and $f^{(n)}(0)=o(n)$ as $n\to\infty$, then $f(1/k)$ is irrational $\forall k\in\mathbb Z\backslash\{0\}$.

Taking, for example, $f(x)=\rho\exp(x)+\mu\sin(x)+\nu\cos(x)$ yields:

Corollary 2. Let $\rho,\mu,\nu$ be integers, not all zero. Then $\rho\exp(1/k)+\mu\cos(1/k)+\nu\sin(1/k)$ is irrational $\forall k\in\mathbb Z\backslash\{0\}$.

In particular, $e\pm\sin 1, e\pm\cos 1,\sin 1\pm\cos 1, e\pm\sin 1\pm\cos 1$  etc are all irrational.