Nth number of continued fraction

Question

Given a real number $r$ and a non-negative integer $n$, is there a way to accurately find the $n^{th}$ (with the integer being the $0^{th}$ number in the continued fraction. If this can not be done for all $r$ what are some specific ones, like $\pi$ or $e$. I already now how to do this square roots.

Answer 1

For arbitrary real numbers, there's no better method known than the 'abstract' one of simply extracting the digits through the usual recurrence relation; even for non-quadratic algebraic numbers — even for a number as simple as $\sqrt{2}+\sqrt{3}$! — nothing is known about any structure to the coefficients.

By contrast, if the goal is to take a number known to some precision and churn out an appropriate number of coefficients for its continued fraction, then there are excellent algorithms for doing that - and that might be enough to find some structure in the coefficients which can then be proven in a non-algorithmic fashion (for instance, the patterns in the continued fraction coefficients of $e$, or of quadratic surds, or in the Liouville numbers). The simplest way is to take a rational approximation $\frac ab$ to your number $r$ (for instance, if you have a decimal expansion $r=d_0.d_1d_2d_3d_4\ldots d_n$ to n digits of precision, then set $a=d_0d_1d_2\ldots d_n$ and $b=10^n$) and then run the extended Euclidean algorithm for the GCD on $a$ and $b$; the 'partial quotients' found along the way are precisely the coefficients of the continued fraction. See http://en.wikipedia.org/wiki/Euclidean_algorithm#Continued_fractions for the basics of the method; if you're interested in more details, volume 2 of Knuth's Art Of Computer Programming (specifically, section 4.5.3, problem 47 within that section and the references there) is an excellent next step.

Answer 2

You can do it recursively: $$\eqalign{f_0(r) &= \lfloor r \rfloor\cr f_{n+1}(r) &= f_n\left( \frac{1}{r - \lfloor r \rfloor}\right)\cr}$$ Of course this may require numerical calculations with very high precision. Actually, if $r$ is a rational number but you don't know it, no finite precision numerical calculation will suffice.