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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.

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.

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Comparison Test

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