Root Test

The central idea

For $\sum a_n$, let $L=\limsup |a_n|^{1/n}$. If $L<1$, the series converges absolutely; if $L>1$, it diverges; if $L=1$, the test is inconclusive. The root test is natural when the full term is raised to the $n$th power.

Definitions, hypotheses, and notation

When the ordinary root limit exists, it equals the limsup and the familiar calculation is enough. The limsup version handles oscillating effective bases. As with the ratio test, $L<1$ yields an eventual geometric bound and therefore absolute convergence.

The root test often simplifies expressions containing $c_n^n$, while factorials usually favor ratios. It can also find power-series radii by producing a factor $|x-a|$. If a remaining root such as $n^{1/n}$ appears, use its limit one rather than treating it as exactly one.

Conceptual meaning

The nth root extracts the effective geometric factor from a term. Polynomial and constant prefactors have nth roots tending to one, leaving the exponential-scale rate that governs the tail.

A dependable method and decision rule

1. Take the absolute value of the general term.
2. Apply the nth root to every factor.
3. Simplify powers before taking the limit or limsup.
4. Compare the resulting $L$ with one.
5. If $L=1$, move to comparison, integral, or another structure-appropriate test.

Fully worked example

Common mistakes and why they fail