Generalized Continued Fraction - Continued Fractions and Series

Continued Fractions and Series

Euler proved the following identity:


a_0 + a_0a_1 + a_0a_1a_2 + \cdots + a_0a_1a_2\cdots a_n =
\frac{a_0}{1-}
\frac{a_1}{1+a_1-}
\frac{a_2}{1+a_2-}\cdots
\frac{a_{n}}{1+a_n}.\,

From this many other results can be derived, such as


\frac{1}{u_1}+
\frac{1}{u_2}+
\frac{1}{u_3}+
\cdots+
\frac{1}{u_n} =
\frac{1}{u_1-}
\frac{u_1^2}{u_1+u_2-}
\frac{u_2^2}{u_2+u_3-}\cdots
\frac{u_{n-1}^2}{u_{n-1}+u_n},\,

and


\frac{1}{a_0} + \frac{x}{a_0a_1} + \frac{x^2}{a_0a_1a_2} + \cdots +
\frac{x^n}{a_0a_1a_2 \ldots a_n} =
\frac{1}{a_0-}
\frac{a_0x}{a_1+x-}
\frac{a_1x}{a_2+x-}\cdots
\frac{a_{n-1}x}{a_n-x}.\,

Euler's formula connecting continued fractions and series is the motivation for the fundamental inequalities, and also the basis of elementary approaches to the convergence problem.

Read more about this topic:  Generalized Continued Fraction

Famous quotes containing the words continued and/or series:

    Calmed by my pleading,
    she continued to list your crimes,
    found that ten fingers weren’t enough
    and cried for a long time.
    Hla Stavhana (c. 50 A.D.)

    If the technology cannot shoulder the entire burden of strategic change, it nevertheless can set into motion a series of dynamics that present an important challenge to imperative control and the industrial division of labor. The more blurred the distinction between what workers know and what managers know, the more fragile and pointless any traditional relationships of domination and subordination between them will become.
    Shoshana Zuboff (b. 1951)