Continued Fraction Expansions

Abstract

A continued fraction [2, 3, 4, 5] is a sequence of fractions

$$ f_n = b_0 + \frac{{a_1 }}{{b_1 + \frac{{a_1 }}{{b_2 + \frac{{a_3 }}{{b_3 + \cdots + \frac{{a_n }}{{b_n }}}}}}}} $$

(3.1)

formed from two sequences a ₁, a ₂,… and b ₀, b ₁,… of numbers. For typographical convenience, definition (3.1) is usually recast as

$$f_n = b_0 + \frac{a_1}{b_1 +} \frac{a_2}{b_2 +} \frac{a_3}{b_3 +} \ldots \frac{a_n}{b_n}.$$

In many practical examples, the approximant f _n converges to a limit, which is typically written as

$$\lim\limits_{n \rightarrow \infty} f_n = b_0 + \frac{a_1}{b_1 +} \frac{a_2}{b_2 +} \frac{a_3}{b_3 +} \ldots.$$

Because the elements a _n and b _n of the two defining sequences can depend on a variable x, continued fractions offer an alternative to power series in expanding functions such as distribution functions. In fact, continued fractions can converge where power series diverge, and where both types of expansions converge, continued fractions often converge faster.