Comparison Test for Divergence

Theorem

Let $\ds \sum_{n \mathop = 1}^\infty b_n$ 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 $\ds \sum_{n \mathop = 1}^\infty a_n$ diverges.

Proof

This is the contrapositive of the Comparison Test.

Hence the result, from the Rule of Transposition.

$\blacksquare$

Also see

• Push Theorem

Sources

• 1992: Larry C. Andrews: Special Functions of Mathematics for Engineers (2nd ed.) ... (previous) ... (next): $\S 1.2.2$: Summary of convergence tests: Theorem $1.3$
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): comparison test
• 2005: Roland E. Larson, Robert P. Hostetler and Bruce H. Edwards: Calculus (8th ed.): $\S 9.4$
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): comparison test