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

Proposition 1.Let be a sequence of non-zero integers such thatexists, and

as .

Then is irrational.

In particular,

Proposition 2.If is a sequence of non-zero integers such that and , thenexists 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 is a unique expansion of the form

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

In this post we shall slightly improve our previous results.

**Proposition 3. **Let and be sequences of non-zero integers such that

exists, and

as .

Then is irrational.

*Proof. *The same argument in the proof of **Proposition 1 **applies.

**Proposition 4.** If and are sequences of non-zero integers such that and then

exists and is irrational.

*Proof. *It suffices to show that and above hold.

Convergence follows easily using the ratio test. So we show that holds.

We have, for sufficiently large ,

as .

So holds, as desired.

As an immediate corollary we get:

**Corollary 1. **Let have an infinite Taylor expansion about that converges for . If and as , then is irrational .

Taking, for example, yields:

**Corollary 2. **Let be integers, not all zero. Then is irrational .

In particular, etc are all irrational.