Introduction to Sequences and Series

Abstract

The first chapter introduces sequences and series and their important properties. Increasing, decreasing, bounded, convergent, and divergent sequences are discussed at an elementary level suitable for an eighth grade student. Arithmetic and geometric progressions are studied in depth and many problems including some Olympiad type problems are given and solved. The famous Fibonacci type sequences are demonstrated and different methods of finding formulae for the nth term of a recursive sequence are given as well as methods of finding recursive formulae for other known series. You will learn interesting methods of finding the nth term and partial sums for series that are not geometric or arithmetic. Figurate numbers (triangular, square, pentagonal, etc) will be introduced from a modern and ancient point of view, in both algebraic and geometric ways. In this chapter you will learn how to prove summation formulae, such as the sum of n natural numbers and the sum of their squares or cubes using different methods including a geometric approach known to ancient Greeks and Babylonians. The reader will gain experience in using sigma notation and, for example, will prove that a cube of a natural number can be written as a sum of precisely n odd consecutive numbers. Each statement is proven and followed by an application problem.