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 *n*th root of the *n*th 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 *n*th 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 affect this ordering.) A series with a “smaller” expression over a “bigger” one converges, for example,

and a series with a “bigger” expression over a “smaller” one diverges, for instance,