Root Test

Summary

The root test uses $L=\underset{n\to \mathrm{\infty }}{lim sup}|a_n{|}^{1/n}$ (or the ordinary limit when it exists). Absolute convergence holds if $L<1$ ; divergence if $L>1$ ; the test is inconclusive if $L=1$ .

Prerequisites

Absolute Convergence, sequences and limits

Theorem

For a series $\sum a_n$ , set

• If $L<1$ , then $\sum a_n$ converges absolutely.
• If $L>1$ (including $L=\mathrm{\infty }$ ), then $a_n\nrightarrow 0$ , so the series diverges.
• If $L=1$ , the test gives no information.

When $lim|a_n{|}^{1/n}$ exists, it equals this limsup.

Conditions / Assumptions

• No sign restriction for the absolute-convergence conclusion.
• Prefer the ratio test when factorials appear; the root test is natural for $n$ th powers.

Worked Example

For $\sum (1/2{)}^n$ , $|a_n{|}^{1/n}=1/2$ , so $L=1/2<1$ : absolute convergence.

For $\sum n^2/e^n$ , $|a_n{|}^{1/n}=n^{2/n}/e\to 1/e<1$ (since $n^{1/n}\to 1$ ), so absolute convergence.

Common Mistakes

• Treating $L=1$ as convergence (harmonic series and $p$ -series both give $L=1$ ).
• Computing $\sqrt[n]{a_n}$ without absolute values when terms change sign.

Connections

• Ratio Test, Power Series (root form of the radius formula)

References

The root test appears in OpenStax Calculus Volume 2.^[1]