Comparison Test for Divergence

Theorem
Let $\displaystyle \sum_{n \mathop = 1}^\infty b_n$ be a divergent series of positive real numbers.

Let $\left \langle {a_n} \right \rangle$ be a sequence in $\R$.

Let:
 * $\forall n \in \N_{>0}: b_n \le a_n$

Then the series $\displaystyle \sum_{n \mathop = 1}^\infty a_n$ diverges.

Proof
This is the contrapositive of the Comparison Test.

Hence the result, from the Rule of Transposition.

Also see

 * Push Theorem