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.