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Infinite Series

9.6 The Ratio and Root Tests

Focus on ratio test, root test, when ratio and root tests are inconclusive.

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Ideas in this lesson

01Ratio TestWorked steps · five questions02Root TestWorked steps · five questions03When Ratio and Root Tests are inconclusiveWorked steps · five questions
Idea 1 of 3

Ratio Test

Compute lim⁡∣an+1an∣\displaystyle \lim |\frac{a_{n+1}}{a_{n}}|. Below one gives absolute convergence, above one gives divergence, and one is inconclusive.

Worked example 1

For an=2nn!(n≥1)a_{n} = \dfrac{2^{n}}{n!} (n \ge 1), find lim⁡∣an+1an∣\displaystyle \lim |\dfrac{a_{n+1}}{a_{n}}|.
  1. ∣an+1an∣=2n+1|\dfrac{a_{n+1}}{a_{n}}| = \dfrac{2}{n + 1}
  2. lim⁡n→∞∣an+1an∣=0\displaystyle \lim\limits_{n\to \infty } |\dfrac{a_{n+1}}{a_{n}}| = 0
  3. L=0<1  ⟹  L = 0 < 1 \implies absolutely convergent

Answer: 00

Worked example 2

For an=n3n(n≥1)a_{n} = \dfrac{n}{3^{n}} (n \ge 1), find lim⁡∣an+1an∣\displaystyle \lim |\dfrac{a_{n+1}}{a_{n}}|.
  1. ∣an+1an∣=n+13n|\dfrac{a_{n+1}}{a_{n}}| = \dfrac{n + 1}{3n}
  2. lim⁡n→∞∣an+1an∣=13≈0.33\displaystyle \lim\limits_{n\to \infty } |\dfrac{a_{n+1}}{a_{n}}| = \dfrac{1}{3} \approx 0.33
  3. L=13≈0.33<1  ⟹  L = \dfrac{1}{3} \approx 0.33 < 1 \implies absolutely convergent

Answer: 13≈0.33\dfrac{1}{3} \approx 0.33

Idea 2 of 3

Root Test

Compute lim⁡ ∣an∣n\displaystyle \lim\,\sqrt[n]{\left|a_{n}\right|}, particularly for terms raised to nn. The same threshold rules apply as in the Ratio Test.

Worked example 1

For an=(2n2(n+1))na_{n} = (\dfrac{2n}{2(n + 1)})^{n}, find the root-test limit.
  1. ∣an∣n=2n2(n+1)\sqrt[n]{\left|a_{n}\right|} = \dfrac{2n}{2(n + 1)}
  2. L=lim⁡n→∞22(1+1n)=1\displaystyle L = \lim\limits_{n\to \infty } \dfrac{2}{2(1 + \dfrac{1}{n})} = 1
  3. L=1  ⟹  L = 1 \implies test inconclusive

Answer: 11

Worked example 2

For an=(3n3(n+1))na_{n} = (\dfrac{3n}{3(n + 1)})^{n}, find the root-test limit.
  1. ∣an∣n=3n3(n+1)\sqrt[n]{\left|a_{n}\right|} = \dfrac{3n}{3(n + 1)}
  2. L=lim⁡n→∞33(1+1n)=1\displaystyle L = \lim\limits_{n\to \infty } \dfrac{3}{3(1 + \dfrac{1}{n})} = 1
  3. L=1  ⟹  L = 1 \implies test inconclusive

Answer: 11

Idea 3 of 3

When Ratio and Root Tests are inconclusive

A test limit of 11 settles nothing. Switch to a pp-Series, Comparison or other appropriate test.

Worked example 1

Apply the Ratio and Root Tests to an=2na_{n} = \dfrac{2}{n}. What do these tests conclude?
  1. ∣an+1an∣=(nn+1)→1|\dfrac{a_{n+1}}{a_{n}}| = (\dfrac{n}{n + 1}) \to 1
  2. ∣an∣n=21nn−1n→1\sqrt[n]{\left|a_{n}\right|} = 2^{\dfrac{1}{n}}n^{-\dfrac{1}{n}} \to 1
  3. L=1  ⟹  L = 1 \implies both tests are inconclusive; the pp-Series Test is needed

Answer: Both are inconclusive

Worked example 2

Apply the Ratio and Root Tests to an=3n2a_{n} = \dfrac{3}{n^{2}}. What do these tests conclude?
  1. ∣an+1an∣=(nn+1)2→1|\dfrac{a_{n+1}}{a_{n}}| = (\dfrac{n}{n + 1})^{2} \to 1
  2. ∣an∣n=31nn−2n→1\sqrt[n]{\left|a_{n}\right|} = 3^{\dfrac{1}{n}}n^{-\dfrac{2}{n}} \to 1
  3. L=1  ⟹  L = 1 \implies both tests are inconclusive; the pp-Series Test is needed

Answer: Both are inconclusive

Resource alignment: Larson & Edwards, Calculus of a Single Variable, Tenth Edition, §9.6,p9.6, p. 627627. Explanations, diagrams and numerical practice variants are original MathLab content aligned to this section.