Continued fractions as inverted recurrence relations

[Max]

On 5/2/06, Don Reble <djr at nk.ca> wrote:

> > Consider a[n] = 2*n - 1 + n^2/a[n+1] with a[1] = 4/Pi.

[...]

>     It may help to observe that a[n]/n decreases, while a[n]/(n+1)
>     increases. Perhaps a[n]/(n+1/2) will be revealing.
>     Using 2000 digits of precision, I find that that quotient
>     reaches 2.41421356 before the accuracy goes. Maybe 1+sqrt(2) is the
>     true limit?

 From the recurrence relation above, it follows that the L=lim a[n]/n
satisfies the following quadratic equation: L^2 - 2*L - 1 = 0 meaning
that L = 1+sqrt(2) or 1-sqrt(2).

Max