CalcRef

Sequences & Series · Convergence Tests

Limit Comparison Test

lim⁡n→∞anbn=L>0⇒∑an and ∑bn both converge or both diverge\lim_{n \to \infty} \frac{a_n}{b_n} = L > 0 \Rightarrow \sum a_n \text{ and } \sum b_n \text{ both converge or both diverge}

If the ratio of terms approaches a positive finite limit, the two series have the same convergence behavior.

  • 2 worked examples
  • 18 in Sequences & Series

Worked examples

Does Σ (2n+1)/(n²+3) converge?
  1. Compare with bₙ = 1/n (dominant terms: 2n/n² = 2/n)
  2. lim aₙ/bₙ = lim n(2n+1)/(n²+3) = lim (2n²+n)/(n²+3) = 2 > 0
  3. Σ 1/n diverges (harmonic), so Σ (2n+1)/(n²+3) diverges.

Answer: Diverges by limit comparison with the harmonic series.

1 more worked examplePremium

Does Σ 1/(2ⁿ - 1) converge?

Step-by-step solution locked.

Unlock the full working. CalcRef Premium opens the step-by-step solution above, and 141 worked solutions across the whole library - the intermediate lines, not just the answer. The formula statement, its conditions, its variables and the first worked example stay free on every page.
See PremiumBrowse the free reference

Related formulas

Comparison Test

See all 18 Sequences & Series formulas on one printable sheet - the same statements and conditions as these pages, typeset and laid out for paper.

Common questions

What is the Limit Comparison Test?

If the ratio of terms approaches a positive finite limit, the two series have the same convergence behavior. It is one of the sequences & series formulas in the CalcRef reference.

How do you use the Limit Comparison Test?

Worked example. Does Σ (2n+1)/(n²+3) converge? Compare with bₙ = 1/n (dominant terms: 2n/n² = 2/n). lim aₙ/bₙ = lim n(2n+1)/(n²+3) = lim (2n²+n)/(n²+3) = 2 > 0. Σ 1/n diverges (harmonic), so Σ (2n+1)/(n²+3) diverges. Answer: Diverges by limit comparison with the harmonic series.

Practice

Quiz yourself on Sequences & Series

Limit Comparison Test is one of 18 Sequences & Series formulas in CalcRef, out of 136 in the library. Today's free round is 10 mixed multiple-choice questions drawn from the same dataset as this page, refreshed every day and replayable as often as you like - no account needed.

Take today's free quizAll 18 Sequences & Series formulas