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Sequences & Series · Convergence Tests

Ratio Test

L=lim⁡n→∞∣an+1an∣:L<1⇒conv.,  L>1⇒div.,  L=1⇒inconclusiveL = \lim_{n \to \infty} \left|\frac{a_{n+1}}{a_n}\right|: \quad L < 1 \Rightarrow \text{conv.},\; L > 1 \Rightarrow \text{div.},\; L = 1 \Rightarrow \text{inconclusive}

The ratio test: compute the limit of the absolute ratio of consecutive terms. Particularly useful for series with factorials or exponentials.

Worked examples

Does Σ n!/nⁿ converge?
  1. |a_{n+1}/aₙ| = [(n+1)!/(n+1)^(n+1)] · [nⁿ/n!] = nⁿ/(n+1)ⁿ = [n/(n+1)]ⁿ
  2. lim [n/(n+1)]ⁿ = lim [1/(1+1/n)]ⁿ = 1/e < 1

Answer: Converges by the ratio test (L = 1/e).

Does Σ 2ⁿ/n! converge?
  1. |a_{n+1}/aₙ| = 2^(n+1)/((n+1)!) · n!/2ⁿ = 2/(n+1)
  2. lim 2/(n+1) = 0 < 1

Answer: Converges by the ratio test.

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