Root Test Root Test

Introduction

For a series,

\sum a_n
:
L = \lim_{n \to \infty} \sqrt[n]{|a_n|} = \lim_{n \to \infty} |a_n|^{1/n}

If
L < 1
, the series converges.
If
L > 1
, the series diverges.
If
L = 1
, the test is inconclusive.

Examples

Example 1:

\sum_{n=1}^{\infty} \left(\frac{1}{2}\right)^n
\begin{align*} a_n &= \left(\frac{1}{2}\right)^n \\ L &= \lim_{n \to \infty} \sqrt[n]{|a_n|} = \lim_{n \to \infty} \sqrt[n]{\left|\left(\frac{1}{2}\right)^n\right|} \\ &= \lim_{n \to \infty} \sqrt[n]{\left(\frac{1}{2}\right)^n} \\ &= \lim_{n \to \infty} \left|\frac{1}{2}\right| \\ &= \frac{1}{2} \end{align*}
Since
L = \frac{1}{2} < 1
, the series
\sum_{n=1}^{\infty} \left(\frac{1}{2}\right)^n
converges by the Root Test.

Example 2:
\sum_{n=1}^{\infty} 2^n
\begin{align*} a_n &= 2^n \\ L &= \lim_{n \to \infty} \sqrt[n]{|a_n|} = \lim_{n \to \infty} \sqrt[n]{|2^n|} \\ &= \lim_{n \to \infty} \sqrt[n]{2^n} \\ &= \lim_{n \to \infty} \left|2\right| \\ &= 2 \end{align*}
Since
L = 2 > 1
, the series
\sum_{n=1}^{\infty} 2^n
diverges by the Root Test.

Example 3:
\sum_{n=1}^{\infty} \frac{1}{n^3}
\begin{align*} a_n &= \frac{1}{n^3} \\ L &= \lim_{n \to \infty} \sqrt[n]{|a_n|} = \lim_{n \to \infty} \sqrt[n]{\left|\frac{1}{n^3}\right|} \\ &= \lim_{n \to \infty} \sqrt[n]{\frac{1}{|n|^3}} \\ &= \lim_{n \to \infty} \frac{1}{|n|^{3/n}} \\ &= 1 \end{align*}
Since
L = 1
, the Root Test is inconclusive for the series
\sum_{n=1}^{\infty} \frac{1}{n^3}
.

We know from the p-series test that this series converges.