## Posts Tagged ‘**continued fractions**’

## The Khinchin-Levy Theorem

To me, one of the most surprising results in number theory is the Khinchin-Levy Theorem. This theorem states that for almost all (in the sense of Lebesgue measure) real numbers , the denominators of the convergents of the continued fraction expansion of satisfy

.

The quantity on the right hand side is the so-called Khinchin-Levy constant. This result is surprising for several reasons:

- Why should the limit even exist?
- Why should it have the same value for almost all ?
- Why should its value involve the three constants , , and ?

Bizarre!

A proof of the Khinchin-Levy Theorem using probability theory can be found in the book *Continued Fractions* by Rockett and Szusz; it can also be proved using the Ergodic Theorem.

## Some more curious continued fractions

We have previously observed that has the remarkably regular continued fraction expansion .

Another curious continued fraction is the following:

,

where the exponents of the 2’s in the continued fraction are the Fibonacci numbers. This unusual continued fraction has been independently rediscovered by several authors. See, for example, Anderson, Brown, and Shiue, *Proc. Amer. Math. Soc.* **123** (1995), 2005-2009.

Among the most famous continued fractions are the Rogers-Ramanujan continued fractions:

.

Ramanujan included these formulas in his famous 1913 letter to Hardy. Later, Hardy wrote, “[These formulas] defeated me completely. I had never seen anything in the least like them before. A single look at them is enough to show that they could only be written down by a mathematician of the highest class. They must be true because, if they were not true, no one would have had the imagination to invent them.”

## The continued fraction expansion of e

It is a somewhat curious fact (due to Euler) that the continued fraction expansion of has a very regular pattern:

.

There are many different proofs of this result. The proof given here is classical and elementary. It can be found, for instance, in the book by Rockett and Szusz.

We begin with the power series expansion of :

.

If is a positive integer, then we have

,

,

.

Let us also define

for , so that and are the series given previously for and , respectively. Our first goal is to show that

.

Expanding the lefthand side, we have

as required. Now let for . We thus have , and furthermore, we have

.

It follows then that is the continued fraction expansion of , and in particular, for , is the continued fraction expansion of .

Recall that if is the -th convergent to the continued fraction , the following hold: , , , , and , for . Thus, if is the the -th convergent of , we have , and .

Let us now recall the claimed expansion . That is, , where , , , and , for . Letting denote the convergents to , we have

,

for . Similarly, we can show that . However, observe that and satisfy the same recurrence as and . Furthermore, when we can check that and ; and similarly, when , we have and . It follows that and for .

We can complete the proof if we can show that converges to as tends to infinity. Note that

.

However, we have previously noted that converges to , so converges to , and the proof is complete.