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

$\text{[math]}$

is convergent, then the limit

$\text{[math]}$

equals zero.

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

If the limit

$\text{[math]}$

is not zero, then the series

$\text{[math]}$

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.