### Introduction

We extend the concept of a finite series, met in Section 16.1, to the situation in which the number of terms increase without bound. We define what is meant by an infinite series being
**
convergent
**
by considering the
**
partial sums
**
of the series. As prime examples of infinite series we examine the harmonic and the alternating harmonic series and show that the former is divergent and the latter is convergent.

We consider various tests for the convergence of series, in particular we introduce the ratio test which is a test applicable to series of positive terms. Finally we define the meaning of the terms absolute and conditional convergence.

#### Prerequisites

- be able to use the $\sum $ summation notation
- be familiar with the properties of limits
- be able to use inequalities

#### Learning Outcomes

- use the alternating series test on infinite series
- use the ratio test on infinite series
- understand the terms absolute and conditional convergence

#### Contents

1 Introduction1.1 Divergence condition for an infinite series

1.2 Divergence of the harmonic series

1.3 Convergence of the alternating harmonic series

2 General tests for convergence

2.1 The alternating series test

3 The ratio test

4 Absolute and conditional convergence