Root Test

Definition

The Root Test is a criterion for determining the absolute convergence of an infinite series by taking the $n$-th root of the absolute value of the $n$-th term as $n$ approaches infinity.

Why It Matters

Like the Ratio Test, the Root Test assesses whether an infinite series is safe and convergent or dangerously divergent. It is particularly essential when a series involves complex powers or exponentials where the ratio test would be computationally messy.

Core Concepts

• The Root Test: Examines the limit: $\rho = \lim_{n \to \infty} \sqrt[n]{|a_n|}$
  • How to read: “The quantity rho equals the limit as n approaches infinity of the n-th root of the absolute value of a n.”
  • Meaning / when to use: Alternative to Ratio Test using root; same decision thresholds. Useful when factors like $(...)^n$ make the ratio test complicated.
    • $\rho < 1$: Absolute convergence.
    • $\rho > 1$: Divergence.
    • $\rho = 1$: Inconclusive.
• Absolute Convergence: Convergence remains even if terms are not all positive.

Connected Concepts

• Ratio Test
• Conditional Tests (Python)
• Comparison Tests for Series