CalcRef

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.

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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