Some help with a continued fraction?

[csprof2000]

I wasn't exactly sure where to put this, so I decided to try this section. It may be more appropriate to put this in the Calculus / Analysis section... I don't know.

Can anybody tell me whether or not it's possible to find the value of the continued fraction

0 + 1 / (2 + 3 / (4 + 5 / (6 + 7 / (...))))

I wrote a relatively simple program to calculate partial fractions... when I go up to the nth odd number, I get this sequence:

1
0.2
0.428571
0.372549
0.38057
0.379654
0.379738
0.379732
0.379732
0.379732
0.379732
0.379732
0.379732
0.379732
0.379732
0.379732
0.379732
0.379732
0.379732
0.379732
0.379732
0.379732
0.379732
0.379732
0.379732

When I go up to the nth even integer, I get:

0
0.5
0.363636
0.381579
0.379562
0.379745
0.379731
0.379732
0.379732
0.379732
0.379732
0.379732
0.379732
0.379732
0.379732
0.379732
0.379732
0.379732
0.379732
0.379732
0.379732
0.379732
0.379732
0.379732
0.379732

So it looks like it converges... to something.

Does anybody know of a way to (a) determine whether it actually converges to anything and (b) find this value analytically, in closed form?

Just a curiosity...

P.S. I know that most continued fractions like this don't have a closed form, I just thought this one was of interest because it seems so simple... just the numbers, you know.

Thanks!

 

[yyat]

According to http://en.wikipedia.org/wiki/Continued_fraction#Regular_patterns_in_continued_fractions" the value can be expressed in terms of modified Bessel functions, also for more general arithmetic series.

 

[csprof2000]

Nice...

 

[CRGreathouse]

There's a comment on Sloane's http://www.research.att.com/~njas/sequences/A113014 that the constant is equal to

sqrt(2e/pi)/erfi(1/sqrt(2)) - 1