## A Gateway to Modern Mathematics Adventures in Iterations II ( Volume 2 ) by Shailesh A Shirali Ramanujan Mathematical Society Little Mathematical Treasures INMO IMO Math Olympiad

This book is a sequel to the author’s A Gateway to Modern Mathematics: Adventures in Iteration, Volume I (which we shall refer to as Adventures 1). The idea of iteration was introduced in that work, together with various associated notions (fixed points, orbits, cycles, limit points, convergence, solution of equations, cobwebbing, and so on), and a large number of examples were studied from the world of arithmetic, algebra and geometry. The present work continues the study of iteration, but at a higher level. However, it is largely self-contained, and can be read without reference to Adventures I.

Students who are preparing for the Regional Mathematics Olympiad (RMO) and the Indian National Mathematics Olympiad (INMO) will find this book useful. It is suitable for self-study by students in the age range 15-21 years (this includes students of mathematics at the college level, who will enjoy seeing “in action some of the facts learnt in their early courses in calculus and analysis), and for lay readers, who will enjoy learning about a topic which is of great current interest. It can also be used by teachers in a school mathematics club.

Preview of book

At the start of the book, in the unnumbered “Review”‘ chapter, we present a very brief review of the material covered in Adventures I.

Following this, in Chapter 1, we examine the insights on iteration provided by differential calculus. Then, in Chapter 2, we study various approaches to the numerical solution of equations (e.g., the Newton-Raphson method). These approaches, as also the iterations studied in Chapters 3 and 4, offer very good examples for illustrating the applications of calculus. In Chapters 5 and 6, we tackle an assorted list of problems. Some of these have their origins in the mathematical olympiads, but we also discuss in detail a well-known problem of Ramanujan’s that was once used in the Putnam Lowell competition, and an open problem related to Mersenne primes. In Chapter 7, we give a brief account of two fascinating discoveries made in recent years the theorems of Li and Yorke, and of Sarkovskii. In Chapter 8. we study the speeds of convergence in iterative processes. Here some knowledge is required of the use of L’Hospital’s rule for computing limits. In Chapter 9, we consider the well-known two-squares theorem of Fermat concerning primes of the form 4n+ 1; we prove it using an iterative technique, and in the process arrive at a constructive proof. In Chapter 10, we study how recursion makes for a simple and clegant construction of certain tree-like structures which have an amazingly “real” appearance. Finally, in.

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