From: petry@accessone.com (David Petry)
Newsgroups: sci.math
Subject: Re: Continued Fraction
Date: Mon, 05 Jan 1998 10:32:28 GMT

>"niteowl" <niteowl@dial.pipex.com> writes:

>>Perhaps someone may have an idea to the proof of the limit of
>>1+              1
>>       --------------------------------------
>>             2+               1
>>                       -------------------
>>                              3+    1
>>                                   -------------
>>                                       4+.....

Here ya' go.

Start with the differential equation

xy'' + y' = y     

and note that

xy''' + 2y'' = y'

and

xy'''' + 3y''' = y''

etc.

For convenience, let  y = y0,  y' = y1, y'' = y2, etc.

Then  y1/y0  =   1/(1 + xy2/y1)   =  1/(1+x/(2 + xy3/y2))  etc.

If we choose the function y to be a solution of the above differential
equation so that  y'(0)/y(0) = 1,   then   we have

y'(x)/y(x) = 1/(1 + x/(2 + x/(3 + ....)))


So y'(1)/y(1)  is the answer you're looking for.