Śleszyński–Pringsheim Theorem
In mathematics, the Śleszyński–Pringsheim theorem is a statement about convergence of certain continued fractions. It was discovered by Ivan Śleszyński and Alfred Pringsheim in the late 19th century. It states that if a_n, b_n, for n=1,2,3,\ldots are real numbers and , b_n, \geq , a_n, +1 for all n, then : \cfrac converges absolutely to a number f satisfying 0<, f, <1, meaning that the series :$f\; =\; \backslash sum\_n\; \backslash left\backslash ,$ where $A\_n\; /\; B\_n$ are the convergents of the continued fraction, converges absolutely.

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Convergent (continued Fraction)
A simple or regular continued fraction is a continued fraction with numerators all equal one, and denominators built from a sequence \ of integer numbers. The sequence can be finite or infinite, resulting in a finite (or terminated) continued fraction like :a_0 + \cfrac or an infinite continued fraction like :a_0 + \cfrac Typically, such a continued fraction is obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer part and another reciprocal, and so on. In the ''finite'' case, the iteration/recursion is stopped after finitely many steps by using an integer in lieu of another continued fraction. In contrast, an ''infinite'' continued fraction is an infinite expression. In either case, all integers in the sequence, other than the first, must be positive. The integers a_i are called the coefficients or terms of the continued fraction. Simple co ...
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