# What is the value of this continued fraction?

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

1. Can we evaluate it ?
2. Is it a nice value ?

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. mathworld.wolfram.com/ContinuedFractionConstant.html Jan 1, 2015 at 7:06
• 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. Jan 1, 2015 at 7:06
• 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. Jan 1, 2015 at 9:35
• @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. Jan 11, 2015 at 2:19
• @turkeyhundt: As Tito Piezas III says, Please convert your comment to an answer. I'l accept it:) Jan 22, 2015 at 9:00

## 1 Answer

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

http://mathworld.wolfram.com/ContinuedFractionConstants.html