# Science:Infinite Series Module/Appendices/The Contrapositive and the Divergence Test

## Introduction

The divergence test is based on the following result that we were able to prove:

 If the series ${\displaystyle \quad \sum _{k=1}^{\infty }a_{k}}$ is convergent, then the limit ${\displaystyle \quad \lim _{k\rightarrow \infty }a_{k}}$ equals zero.

We claimed that it is equivalent to this statement (which is the divergence test):

 If the limit ${\displaystyle \lim _{k\rightarrow \infty }a_{k}}$ is not zero, then the series ${\displaystyle \sum _{k=1}^{\infty }a_{k}}$ is not convergent.

Let's look at this more closely to see why this would be the case. We will use the concept of the contrapositive.

## Topics

1. Definition of the Contrapositive
2. A Few Contrapositive Examples
3. A Closer Look at the Contrapositive with Sets