Dr. Ritika
Engineering
December 2022
A continuous fraction is a mathematical expression formed by expressing a number as the sum of its integer component and the reciprocal of another number, then expressing this other number as the sum of its integer component and yet another reciprocal, etc. This procedure is repeated until the desired result is achieved. When an integer is substituted with another continued fraction, the iteration or recursion results in a continued fraction with a limited number of steps. Many standard functions, such as the arctangent and the logarithm, can have the tails of the Taylor series written as continuing fractions. Some of these continuous fractions have the startling side effect of dramatically accelerating the convergence of the underlying power series. This inquiry was prompted by an unexpected observation on Gregory's series. Gregory's series. Exist are functions that can be evaluated analytically and expressed in hypergeometric form. Ramanujan's discoveries on continuing fractions are clear consequences of relationships between three-term hypergeometric series. Theirq-analogues result in a number of the continuing fractions detailed in the
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