2. Further Study of Sequences and Series

Chapter 2 continues the study of sequences and series. Although the ancient Greeks and Babylonians knew how to add integers and proved many formulae using geometry, there were limitations to their techniques. For example, they did not know how to represent the fourth power of a number because we all live in a three-dimensional space. Additionally, there are infinitely many examples of sequences and series that involve fractional expressions or combinations of exponential and trigonometric functions that were unknown in ancient times. In Chapter 2 you will learn different methods of finding exact sums and will prove many formulae using other creative ideas. It introduces the Leibniz harmonic triangle and shows its connection with the well-known Pascal triangle. Properties of the Leibniz triangle will be used to prove sums of infinite series. Additionally, trigonometric series are introduced and some interesting methods of evaluating finite and infinite trigonometric series. While there is no formula for a sequence of prime numbers, using the Cantor theorem, the reader will attempt to find a finite arithmetic progression that contains only prime numbers. Methods of proofs are mastered in this chapter.