Ratio Test

Definition

The Ratio Test is a criterion for determining the absolute convergence of an infinite series by measuring the ratio of successive terms as $n$ approaches infinity.

Why It Matters

The Ratio Test is the “gatekeeper” of series stability, especially for power series. In engineering and physics, if you sum a divergent series, your results will explode toward infinity. This test lets us know with certainty whether an infinite process will sum to a finite, safe number.

Core Concepts

• The Ratio Test: Examines the limit: $\rho = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|$
  • How to read: “The quantity rho equals the limit as n approaches infinity of the absolute value of a n plus one divided by a n.”
  • Meaning / when to use: If rho < 1 the series converges absolutely; >1 diverges; =1 inconclusive. It is the workhorse test for finding the radius of convergence for power series.
    • $\rho < 1$: Absolute convergence.
    • $\rho > 1$: Divergence.
    • $\rho = 1$: Inconclusive.
• Absolute Convergence: Convergence remains even if terms are not all positive.

Connected Concepts

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