Rational Convergents to General and Regular Continued Fractions

RamaWolf

Let a General Continued Fraction (gCF) be given as

(1) b[itex]_{0}[/itex]+[itex]\frac{a_{1}}{b_{1}+\frac{a_{2}}{b_{2}+\frac{a_{3}}{{b_{3}+...}}}}[/itex]

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

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

A[itex]_{n}[/itex] := b[itex]_{n}[/itex]*A[itex]_{n-1}[/itex] + a[itex]_{n}[/itex]*A[itex]_{n-2}[/itex] and B[itex]_{n}[/itex] := b[itex]_{n}[/itex]*B[itex]_{n-1}[/itex] + a[itex]_{n}[/itex]*B[itex]_{n-2}[/itex]

(A[itex]_{-1}[/itex] := 1, A[itex]_{0}[/itex] := b[itex]_{0}[/itex], B[itex]_{-1}[/itex] := 0, B[itex]_{0}[/itex] := 1)

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

(2) b[itex]_{0}[/itex]+[itex]\frac{1}{b_{1}+\frac{1}{b_{2}+\frac{1}{{b_{3}+...}}}}[/itex]

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

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

A[itex]_{n}[/itex] := b[itex]_{n}[/itex]*A[itex]_{n-1}[/itex] + A[itex]_{n-2}[/itex] and B[itex]_{n}[/itex] := b[itex]_{n}[/itex]*B[itex]_{n-1}[/itex] + B[itex]_{n-2}[/itex]

(A[itex]_{-1}[/itex] := 1, A[itex]_{0}[/itex] := b[itex]_{0}[/itex], B[itex]_{-1}[/itex] := 0, B[itex]_{0}[/itex] := 1)