Root Criterion

analysis

Definition

Root Criterion

The root criterion (or root test) is a convergence criterion for an infinite series ${\sum }_{n=0}^{\infty }{a}_n$:

1. If there exists $q<1$ such that $\sqrt[n]{|a_n|}\le q$ for almost all $n$, then the series is absolutely convergent.
2. If $\sqrt[n]{|a_n|}\ge 1$ for infinitely many $n$, then the series is divergent.

Limit Form

Using the limit superior:

• ${\mathrm{limsup}}_{n\to \infty }\sqrt[n]{|a_n|}<1⟹$ absolutely convergent.
• ${\mathrm{limsup}}_{n\to \infty }\sqrt[n]{|a_n|}>1⟹$ divergent.

Significance

The root criterion is generally more powerful than the Quotient Criterion, as every series that is shown to converge by the quotient criterion will also be shown to converge by the root criterion, but the converse is not true.