Definition of the number e

This chapter contains the beginnings of the most important, and probably the most subtle, notion in mathematical analysis,i.e., the concept of a limit. Though Newton and Leibniz discovered the calculus with itstangent lines described as limits of secant lines, and though the Greeks were already estimating areas of regions by a kind of limiting process,the precise notion of limit that we use today was not formulated until the 19th century by Cauchy and Weierstrass.

The main results of this chapter are the following:

1. The definition of the limit of a sequence ,
2. The definition of the real number $e$ ( [link] ),
3. The Squeeze Theorem ( [link] ),
4. the Bolzano Weierstrass Theorem ( [link] and [link] ),
5. The Cauchy Criterion ( [link] ),
6. the definition of an infinite series ,
7. the Comparison Test ( [link] ), and
8. the Alternating Series Test ( [link] ).

These are powerful basic results about limits that will serve us well in later chapters.