How to Use the Root Test to Determine Whether a Series Converges
The root test doesn’t compare a new series to a known benchmark
series. It works by looking only at the nature of the series
you’re trying to figure out. You use the root test to
investigate the limit of the nth root of the nth term of your
series. Like with the ratio test, if the limit is less than 1,
the series converges; if it’s more than 1 (including infinity),
the series diverges; and if the limit equals 1, you learn
nothing. The root test is a good one to try if the series
involves nth powers. Try this one: Does converge or diverge?
Here’s what you do: Because the limit is less than 1, the series
converges. Sometimes it’s useful to make an educated guess about
the convergence or divergence of a series before you launch into
one or more of the convergence/divergence tests. Here’s a tip
that helps with some series. The following expressions are
listed from smallest to biggest: (The 10 is an arbitrary number;
the size of the number doesn’t af