The Root test is used as follows.
Algorithm:
We are given a seriesak with non-negative terms.
Step 1. Evaluate the limitStep 2. Assume that this limit exists.
• Ifϱ < 1, then the given series converges.
• Ifϱ > 1, then the given series diverges.
This test is obviously perfect for series whose term are of the form
If p(x) is a non-zero polynomial, then
Note, however, that the case
Example: Decide on convergence of the series
The denominator just calls for the Root test. Using the fact above we get
Since
For other examples see especially this problem, this problem, and this problem in Solved Problems - Testing convergence.
Sometimes one has to use more general (and less convenient) versions of the Root test, see Root and Ratio test in Theory - Testing convergence; for an example see this problem in Solved Problems - Testing convergence, also look at this problem.
One can also use this test to get some information about series with general terms, not just non-negative, see the Note at the end of the section Root and Ratio test in Theory - Testing convergence.