An interesting serendipitous number

 Last seek I blogged about two math problems of interest to me here.

One of them two people posted answers, which was great since I didn't know how to solve them and now I do. Yeah! I blogged about that here.

The other problem got no comments, so I suppose it was of interest to me but not others. I was interested in it because the story behind it is interesting, and the answer is interesting.

it is from the paper 

An interesting and serendipitous number by John Ewing and Ciprian Foias, which is a chapter in the wonderful book 

Finite vs Infinite: Contributions to an eternal dilemma

Here is the story, I paraphrase the article (I'll give pointers  later).

In the mid 1970's a student asked Ciprian about the following math-competition problem:

x(1)>0    x(n+1) =  (1 + (1/x(n)))^n. For which x(1) does x(n) --> infinity?

It turned out this was a misprint. The actual problem was

x(1)>0  x(n+1)=(1+(1/x(n))^{x(n)}. For which x(1) does x(n) --> infinity.

The actual math-comp problem  (with exp x(n)) is fairly easy (I leave it to you.) But this left the misprinted problem (with exp n).  Crispian proved that there is exactly ONE x(1) such that x(n)--> infinity. 

Its approx 1.187... and may be trans.

I find the story and the result interesting, but the proof is to long for a blog post.

I tried to find the article online and could not. A colleague found the following:

A preview of the start of the article here

Wikipedia Page on the that number, called the Foias constant, here

Mathworld page on that number here

Most of the article but skips two pages here