Ratio test: Methods survey

The Ratio test is used as follows.

Algorithm:
We are given a series  ak  with positive terms.
Step 1. Evaluate the limit

Step 2. Assume that this limit exists.
• If λ < 1, then the given series converges.
• If λ > 1, then the given series diverges.

This test is well-suited to series for which the ratio of successive terms is easy to handle, which above all means series that feature factorials; terms of the type qk are also fine. Polynomials can be handled quite well, but they contribute 1:

If p(x) is a non-zero polynomial, then

Since the case λ = 1 is indecisive, polynomial parts are not helpful when deciding convergence in the Ratio test; in order to reach a conclusion, the given series must have also parts of different type (like ck ).

Example: Decide on convergence of the series

The denominator just calls for the Ratio test. We get

Since λ < 1, the given series converges.

Note that when evaluating the limit for lambda, we split it into two limits. If the expression for ak consists of several parts, then it is usually best to gather the corresponding entries together when doing the Ratio test ("each to its sibling"), since when two terms coming from the same basic source are compared, then we have the best chance to simplify. We also saw why factorials just love the Ratio test.

For other examples see this problem, this problem, this problem, this problem, this problem, and this problem in Solved Problems - Testing convergence.

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. You find also information about more general versions of this tests there.


Integral test
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