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.

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