Comparison Test for Divergence
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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
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