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 $\sum _{k=1}^{\mathrm{\infty }}{a}_k$ is convergent, then the limit $\underset{k\to \mathrm{\infty }}{\lim }{a}_k$ equals zero.

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

If the limit $\underset{k\to \mathrm{\infty }}{\lim }{a}_k$ is not zero, then the series $\sum _{k=1}^{\mathrm{\infty }}{a}_k$ 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