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.

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