# Infinite Series Expansions - Introduction

By M. Bourne

Ever wondered:

- How a calculator can give you the value of sine (or cos, or cot) of
*any*angle? - How it can give you the square root (or cube root, or 4th root) of
*any*positive number? - How it can find the logarithm of
*any*(positive) number you give it?

Does a calculator store every answer that every human may ever ask it?

Actually, no. The pocket calculator just remembers special polynomials and substitutes whatever you give it into that polynomial. It keeps substituting into terms of that polynomial until it reaches the required number of decimal places. It then displays the answer on the screen. [See the definition of a polynomial.]

So what are these special polynomials? In this chapter, we learn that they are **infinite series expansions**. We also learn how to obtain such expansions, using differentiation.

## Why Study Series Expansions?

### Revision Tip

Need some background information? See Geometric Progressions and Infinite Geometric Series from an earlier chapter.

Apart from their uses in calculators and computers, infinite series expansions are used:

- To find approximations for special numbers like π and
*e*. - To simplify some mathematical processes. We can differentiate (or integrate) term-by-term in our resulting series.
- In many mathematical proofs (we can solve certain theorems by expanding out the functions involved and examining what goes on.)

This section also serves as an introduction to the Fourier Series, that we meet later.

## In this Chapter

1. Taylor Series - which can express a function in polynomial form, near some local value.

2. Maclaurin Series - which expands the function near *x* = 0

3. How Does a Calculator Work? - which explains how calculators find values of functions