What is the value of this continued fraction?

Question

I am curious about the value of the continued fraction $$1+\cfrac{1}{2+\cfrac{1}{3+\cfrac{1}{4+\cfrac{1}{5+\cfrac{1}{6+\dots}}}}}.$$

Clearly it should be a transcendental number. But I have no idea about calculate it.

Here is some info that is a little beyond my comprehension at this hour. http://mathworld.wolfram.com/ContinuedFractionConstant.html — turkeyhundt, Jan 01 '15 at 07:06
It has a quite surprising (at least to me) nice value as a ratio of Bessel functions: $\frac{I_{0}{(2)}}{I_{1}{(2)}}$. I don't have any idea how to demonstrate this though. See WRA. — David H, Jan 01 '15 at 07:06
I thought people newer thought about this continued fraction. But I was wrong :) I found a similar continued fraction which may be related to this one. — Bumblebee, Jan 01 '15 at 09:35
@turkeyhundt: Rather than this question be tagged as "unanswered", can you modify your comment to an answer? After all, you found: 1) its name, 2) the link provides the evaluation, and 3) it seems to have a "nice" value. — Tito Piezas III, Jan 11 '15 at 02:19
@turkeyhundt: As Tito Piezas III says, Please convert your comment to an answer. I'l accept it:) — Bumblebee, Jan 22 '15 at 09:00

score 2 · Accepted Answer · edited Dec 14 '19 at 18:00

2

Here's the info on this continued fraction and others.

edited Dec 14 '19 at 18:00

Tiago André

answered Jan 22 '15 at 09:02

turkeyhundt

1 Answers1