Ratio Test

Q: How do you use the Ratio Test?

Worked example. Does Σ n!/nⁿ converge? |a_{n+1}/aₙ| = [(n+1)!/(n+1)^(n+1)] · [nⁿ/n!] = nⁿ/(n+1)ⁿ = [n/(n+1)]ⁿ. lim [n/(n+1)]ⁿ = lim [1/(1+1/n)]ⁿ = 1/e < 1. Answer: Converges by the ratio test (L = 1/e).

L = \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.

2 worked examples
18 in Sequences & Series

Worked examples

Does Σ n!/nⁿ converge?

|a_{n+1}/aₙ| = [(n+1)!/(n+1)^(n+1)] · [nⁿ/n!] = nⁿ/(n+1)ⁿ = [n/(n+1)]ⁿ
lim [n/(n+1)]ⁿ = lim [1/(1+1/n)]ⁿ = 1/e < 1

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

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Does Σ 2ⁿ/n! converge?

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Common questions

What is the Ratio Test?

The ratio test: compute the limit of the absolute ratio of consecutive terms. Particularly useful for series with factorials or exponentials. It is one of the sequences & series formulas in the CalcRef reference.

How do you use the Ratio Test?

Worked example. Does Σ n!/nⁿ converge? |a_{n+1}/aₙ| = [(n+1)!/(n+1)^(n+1)] · [nⁿ/n!] = nⁿ/(n+1)ⁿ = [n/(n+1)]ⁿ. lim [n/(n+1)]ⁿ = lim [1/(1+1/n)]ⁿ = 1/e < 1. Answer: Converges by the ratio test (L = 1/e).

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