Comparison Tests for Series

Definition

Comparison tests are a suite of methods used to determine the convergence or divergence of a series $\sum a_n$ with non-negative terms by comparing it to a “benchmark” series $\sum b_n$ whose behavior is already known. They rely on the order-preserving properties of summation.

• How to read: “Sum of a n compared to sum of b n.”
• Meaning: If terms of $a_n$ are controlled by known $b_n$, inherit convergence or divergence from the benchmark.

Why It Matters

Comparison tests are the ‘benchmarking tool’ of calculus; they allow us to determine the stability of an infinite system by comparing it to a known quantity, a critical skill for ensuring convergence in physical models.

Core Concepts

• Direct Comparison Test (DCT):

  • If $0 \leq a_n \leq b_n$ and $\sum b_n$ converges, then $\sum a_n$ must converge.
  • If $0 \leq b_n \leq a_n$ and $\sum b_n$ diverges, then $\sum a_n$ must diverge.
  • How to read: “If a n is between zero and b n, and the sum of b converges, then the sum of a converges.”
  • Meaning / when to use: Smaller-than-convergent converges; larger-than-divergent diverges. Sandwich the unknown between known benchmarks.

• Limit Comparison Test (LCT): If $a_n, b_n > 0$ and $\lim_{n \to \infty} \frac{a_n}{b_n} = c \in (0, \infty)$, then both series share the same fate.

  • How to read: “The limit as n approaches infinity of the ratio of a n to b n equals c, a positive finite number.”
  • Meaning / when to use: Asymptotically equivalent terms—if the ratio settles to a positive constant, both series converge or both diverge.

• Benchmarks: Commonly used benchmark series include $p$-series and geometric series.

Connected Concepts

• Geometric Sequence and Geometric Series
• Conditional Tests (Python)
• Comparison Properties of Integrals