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10.6 Comparison Tests for Convergence

Definition: Direct Comparison Test

Suppose \(0 \le a_n \le b_n\) for all \(n\):

  1. If the "larger" series \(\sum b_n\) converges, then the "smaller" series \(\sum a_n\) converges.

  2. If the "smaller" series \(\sum a_n\) diverges, then the "larger" series \(\sum b_n\) diverges.

Definition: Limit Comparison Test

Suppose \(a_n > 0\) and \(b_n > 0\).

If \(\lim_{n \to \infty} \frac{a_n}{b_n} = L\), where \(0 < L < \infty\), then both series \(\sum a_n\) and \(\sum b_n\) share the same behavior (both converge or both diverge).