5.14 Root test
Root test
Lessons
Notes:
Note *Let $\sum a_n$ be a positive series. Then we say that
$R=$ $\lim$_{n →$\infty$} $\mid a_n\mid^{\frac{1}{n}}$
Where:
1. If $R$ < $1$, then the series is convergent (or absolutely convergent)
2. If $R$ > $1$, then the series is divergent
3. If $R=1$, then the series could either be divergent, or convergent
Basically if $R=1$, then the root test fails and would require a different test to determine the convergence or divergence of the series.
Note that if the root test gives $R=1$, then so will the ratio test.

3.
Convergence & Divergence of Root test
Use the Root test to determine if the series converges or diverges. If the root test does not determine the convergence or divergence of the series, then resort to another test.