Applications of Series in Calculus: Approximating Radical Functions Through Rational Expressions

Through a minor adaptation of a result that is outlined in a work on infinite series by Konrad Knopp, the natural logarithm function can be represented by an infinite series of rational expressions that converges over the function's domain. In this article, we illustrate how this approach can be utilized to represent a radical function, allowing for its approximation by ratios of first degree polynomials. We describe how these ratios may yield more accurate numerical estimates than the partial sums of the corresponding series obtained from the Binomial Theorem while requiring fewer computations. This material would be suitable as an enrichment topic in second-semester calculus courses.