Ratio and Root Tests

Key idea

Convergence via limit of aₙ₊₁/aₙ.

In depth

For L = lim |aₙ₊₁/aₙ| (ratio) or lim |aₙ|^(1/n) (root): if L < 1 the series converges absolutely, if L > 1 it diverges, if L = 1 the test is inconclusive. Good for factorials and powers.

Worked example

Σ n!/10ⁿ: ratio = (n + 1)/10 → ∞, so it diverges. Σ 2ⁿ/n!: ratio = 2/(n + 1) → 0, converges.

Formula

Real-world uses

• Checking convergence of series used in physics and in the formulas for special functions

Common mistake

Using it on p-series, where L = 1 always.

How to learn this

1. Read the key idea and say it in your own words.
2. Work through the worked example, then try a similar problem.
3. Use the interactive lesson’s visual and mini-quiz to check the pattern.

Frequently asked questions

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Related expert topics in Calculus

• Derivatives of Inverse Trig – Derivatives of arcsin, arccos, arctan.
• Implicit Differentiation – Differentiate both sides of F(x,y)=0 treating y as a function of x.
• Logarithmic Differentiation – Take logs before differentiating products and powers.
• Related Rates – Link rates of change of connected quantities.
• Linear Approximation – Approximate f near a using its tangent line.
• Differentials – Use dy = f′(x)dx for small changes.