Abstract:
The convergence of continued fractions is defined in a manner other than the conventional definition. A new summation method is used to determine the values of continued fractions and series that diverge in the classical sense. The method is applicable not only to ordinary continued fractions, but also to ones of other classes, for example, to Hessenberg continued fractions. As a result, an original algorithm for finding zeros of $n$th-degree polynomials is constructed. The $r/\varphi$-algorithm proposed is also used to solve infinite systems of linear algebraic equations.
Key words:high-degree algebraic equations, divergent continued fractions, infinite systems of linear algebraic equations, summation algorithm for divergent continued fractions.
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