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Sequences & Series · Convergence Tests

Comparison Test

0≤an≤bn:∑bn conv.⇒∑an conv.;∑an div.⇒∑bn div.0 \leq a_n \leq b_n: \sum b_n \text{ conv.} \Rightarrow \sum a_n \text{ conv.}; \quad \sum a_n \text{ div.} \Rightarrow \sum b_n \text{ div.}

Compare with a known series. If the larger series converges, so does the smaller. If the smaller diverges, so does the larger.

  • 2 worked examples
  • 18 in Sequences & Series

Worked examples

Does Σ 1/(n³ + 1) converge?
  1. 1/(n³+1) < 1/n³ for all n ≥ 1
  2. Σ 1/n³ is a p-series with p = 3 > 1 (converges)
  3. By comparison, Σ 1/(n³+1) converges.

Answer: Converges by comparison with Σ 1/n³.

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Does Σ (ln n)/n from n = 3 converge?

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Related formulas

Limit Comparison Test

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Common questions

What is the Comparison Test?

Compare with a known series. If the larger series converges, so does the smaller. If the smaller diverges, so does the larger. It is one of the sequences & series formulas in the CalcRef reference.

How do you use the Comparison Test?

Worked example. Does Σ 1/(n³ + 1) converge? 1/(n³+1) < 1/n³ for all n ≥ 1. Σ 1/n³ is a p-series with p = 3 > 1 (converges). By comparison, Σ 1/(n³+1) converges. Answer: Converges by comparison with Σ 1/n³.

Practice

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