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

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