Infinite Series

so π is an “infinite sum” of fractions. Decimal expansions like this show that an infinite series is not a paradoxical idea, although it may not be clear how to deal with non-decimal infinite series like (1.1) at the moment. Infinite series provide two conceptual insights into the nature of the basic functions met in high school (rational functions, trigonometric and inverse trigonometric functions, exponential and logarithmic functions). First of all, these functions can be expressed in terms of infinite series, and in this way all these functions can be approximated by polynomials, which are the simplest kinds of functions. That simpler functions can be used as approximations to more complicated functions lies behind the method which calculators and computers use to calculate approximate values of functions. The second insight we will have using infinite series is the close relationship between functions which seem at first to be quite different, such as exponential and trigonometric functions. Two other applications we will meet are a proof by calculus that there are infinitely many primes and a proof that e is irrational.