By Nicholas D. Kazarinoff

Applicable for undergraduates or even highschool scholars, this article introduces a couple of historical theorems, explores inequalities and calculus, and covers glossy theorems, together with Bernstein's facts of the Weierstrass approximation theorem and the Cauchy, Bunyakovskii, Hölder, and Minkowski inequalities. 1961 variation. comprises 28 figures.

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**Example text**

It may be surprising that the approximation of square-roots should result in such image, but in fact throughout the history of science and mathematics even the simple-looking task of approximating square-root of numbers has resulted in remarkable discoveries. This task was also the inspiration behind polynomiography. Later in the chapter we will extend our algebraic approach to the case of approximating cube-root of a given positive number α, corresponding to the real root of p(z) = z 3 − α. The general case of polynomials will be treated thoroughly throughout the book.

Thus if for each j = 0, . . , m−1 the corresponding subsequences converge to θ, so will the entire sequence of {zk }∞ k=0 . We now prove the subsequence {N km+j (z0 )}∞ k=0 converges to θ for all j = 0, 1, . . , m − 1. For j = 0 the subsequence corresponds to the fixed point iterates of N m , hence convergent to θ by assumption. To prove that it is true for j = 1, we note N km+1 (z0 ) = N (N mk (z0 )). Since the sequence of N mk (z0 ) converges to θ, continuity of N at θ implies the sequence of images is convergent to N (θ).

Chapter 21, offers further applications and extensions of the Basic Family and polynomiography. Whenever possible the book complements the concepts within each chapter via polynomiography images. Acknowledgements. I would like to thank many people who in one form or another - have been helpful in the course of completion of this book over several years, or with respect to polynomiography activities addressed in this book, or with respect to the motivation of these activities, or for participation in these activities, or simply for their encouragements.