Comparison Test for Divergence

Theorem

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

Let $\sequence {a_n}$ 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.

$\blacksquare$