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