Mathfolis

Ratio & Root Tests

Unit 10 · Infinite Sequences & Series

Ratio Test and Absolute/Conditional Convergence

The Ratio Test computes L = lim|aₙ₊₁/aₙ|. If L < 1, the series converges absolutely; if L > 1 (or ∞), it diverges; if L = 1, the test is inconclusive. It is most effective when the series involves factorials or exponentials. A series is absolutely convergent if ∑|aₙ| converges, conditionally convergent if ∑aₙ converges but ∑|aₙ| diverges.

Ratio Test

L=lim⁡n→∞∣an+1an∣L = \lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|
L < 1: converges absolutely | L > 1: diverges | L = 1: inconclusive

Absolute vs. Conditional Convergence

Absolutely convergent: ∑|aₙ| converges
Conditionally convergent: ∑aₙ converges but ∑|aₙ| diverges
AP Tip: Use the Ratio Test when the series has n! or expressions of the form aⁿ. For (n+1)!/n!, remember (n+1)! = (n+1)·n!.
Caution: The Ratio Test gives L = 1 for p-series — it is inconclusive there. Use the p-series rule instead.
Type 1

Ratio Test with Factorials

Compute aₙ₊₁/aₙ, simplify using (n+1)! = (n+1)·n!, then take the limit.

Example 1

Determine whether ∑n!/3ⁿ converges or diverges.

Example 2

Determine whether ∑3ⁿ/n! converges or diverges.

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Type 2

Ratio Test with Exponentials

For series with nᵏ/aⁿ form, the ratio simplifies to ((n+1)/n)ᵏ · (1/a), whose limit gives a/a ratio.

Example 3

Determine whether ∑n²/2ⁿ converges or diverges.

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Type 3

Absolute vs. Conditional Convergence

First test ∑|aₙ| (remove the alternating sign). If it converges: absolutely convergent. If it diverges but ∑aₙ converges (by AST): conditionally convergent.

Example 4

Classify ∑(−1)ⁿ/√n as absolutely convergent, conditionally convergent, or divergent.

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