# Rational Convergents to General and Regular Continued Fractions

1. Jul 5, 2011

### RamaWolf

Let a General Continued Fraction (gCF) be given as

(1) b$_{0}$+$\frac{a_{1}}{b_{1}+\frac{a_{2}}{b_{2}+\frac{a_{3}}{{b_{3}+...}}}}$

or abbreviated as b$_{0}$+$\frac{a_{1}}{b_{1}+}$ $\frac{a_{2}}{b_{2}+}$ $\frac{a_{3}}{b_{3}+}$ $\frac{a_{4}}{b_{4}+}$ $\frac{a_{5}}{b_{5}+}$ $\frac{a_{6}}{b_{6}+}$ $\frac{a_{7}}{b_{7}+}$ $\frac{a_{8}}{b_{8}+}$ ...

then n-th rational convergent to (1) is C$_{n}$ := $A_{n}/B_{n}$ with

A$_{n}$ := b$_{n}$*A$_{n-1}$ + a$_{n}$*A$_{n-2}$ and B$_{n}$ := b$_{n}$*B$_{n-1}$ + a$_{n}$*B$_{n-2}$

(A$_{-1}$ := 1, A$_{0}$ := b$_{0}$, B$_{-1}$ := 0, B$_{0}$ := 1)

For a Regular Continued Fraction (rCF) all a$_{i}$ are set to +1, giving

(2) b$_{0}$+$\frac{1}{b_{1}+\frac{1}{b_{2}+\frac{1}{{b_{3}+...}}}}$

or abbreviated [b$_{0}$; b$_{1}$,b$_{2}$,b$_{3}$,b$_{4}$,b$_{5}$,...]

and the formula for the n-th rational convergents to (2) C$_{n}$ := $A_{n}/B_{n}$ with

A$_{n}$ := b$_{n}$*A$_{n-1}$ + A$_{n-2}$ and B$_{n}$ := b$_{n}$*B$_{n-1}$ + B$_{n-2}$

(A$_{-1}$ := 1, A$_{0}$ := b$_{0}$, B$_{-1}$ := 0, B$_{0}$ := 1)