ContinuedFractionK

The continued fraction $f_1/(g_1 + f_2/(g_2 + \cdots))$ over an index range, finite or infinite.

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ContinuedFractionK(f, g, (i, imin, imax))the finite continued fraction, exact, for imin ≤ i ≤ imax

ContinuedFractionK(f, g, (i, imin, PositiveInfinity))the infinite fraction, when

f

and

g

do not depend on

i

Typed signatures

Type	Package	Applies
`(any, any, tuple<symbol, integer, any>) -> number`	`modular`	always

Domain: The modular group

Details

The iterator is written the same way Sum and Product write theirs: Tuple(i, imin, imax)
The finite fraction is built right to left: $f_{imax}/g_{imax}$ first, then each $f_i/(g_i + \text{that})$ down to $i = imin$
The infinite case is solved algebraically, not truncated: constant $f, g$ make $x = f/(g+x)$ , whose positive root $x = (-g + \sqrt{g^2+4f})/2$ is the fraction's value — an $f$ or $g$ that depends on $i$ has no such closed form here, and the call stays unevaluated

Examples

Basic #

$\cfrac{1}{1 + \cfrac{1}{2 + \cfrac{1}{3 + \cfrac{1}{4 + \cfrac15}}}}$

\mathrm{ContinuedFractionK}(1, k, (k,1,5))

Out

\frac{157}{225}

Scope #

the infinite all-ones fraction is $1/\varphi$ — $(\sqrt5 - 1)/2$

\mathrm{ContinuedFractionK}(1, 1, (k,1,∞))

Out

\frac{\sqrt{5}-1}{2}

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