Section 10.11

The Root Test

A specialized test for series involving -th powers.

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Introduction

When a series has terms like , the Root Test is often the fastest way to check for convergence. It works by eliminating the -th power.

Interactive: The nth Root Limit

Green: Root limit (Converges).
Red: Root limit (Diverges).

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The Root Test

Definition

Let be a series. Define:
  • If : Abs. Convergent.
  • If : Divergent.
  • If : Inconclusive.

Useful Fact

.
This comes up often when taking the n-th root of polynomials like or .

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Worked Examples

Example 1: Classic Case (Level 1)

Test .

Apply Root Test:

Limit of rational function (leading coeffs):
.

Since , the series Converges Absolutely.

Example 2: With Factorial (Tricky) (Level 3)

Test . (Wait, this is easier with Root than Ratio!)

Rewrite as .
.

Converges ().

Contrast with vs . Ratio test usually handles better, but screams Root Test.

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Practice Quiz

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