Testing Convergence of Series: Methods Survey

If you wish to simultaneously follow another text on testing convergence in a separate window, click here for Theory and here for Solved Problems.

Testing convergence of series (of real/complex/... numbers and of functions) is very important and at its heart one finds tests for series of non-negative terms, which is the main topic of this section. After that we look at testing convergence of series whose signs change, look at general series, then pass to absolute convergence, and at the end we summarize investigating (classifying) convergence.

While knowing mechanics of various tests is obviously necessary, equally important is the art of choosing the right test. While there are many criteria of various strength and usefulness (see e.g. Theory - Testing convergence), here we focus on tests that are usually covered in introductory courses.

Testing convergence of series with non-negative terms.

We are given a series  ∑ a_k  of real numbers satisfying a_k ≥ 0. How do we recognize its convergence?

• If we can readily show that the terms a_k of the series do not tend as a sequence to 0, then we know that the series diverges. With many series one can see at the first glance whether their terms go to zero or not, then it is worth to start with this test, since it does not cost us anything. However, when the problem of convergence of terms looks complicated, then it is not clear whether such investigation pays off, since this test often fails and then we just lost time. Then I prefer to start with other tests (if there are some suitable ones) and leave this one as the last resort.
• If the terms of the series consist of polynomials (typically if they are given by a rational function), then we can usually determine its convergence using the comparison tests.
  In general, if a series can be simplified by ignoring less important parts in it, then comparison tests are a good start.
  Note: Using comparison properly requires in most cases good knowledge of p-series.
• If the terms of the series include powers of the type c^k or generally terms of the type (c_k)^k, then we can often determine its convergence using the Root test.
• If the terms of the series include factorials and/or powers of the type c^k, then we can often determine its convergence using the Ratio test.
  Note that if a series features factorials then the Ratio test is usually the test of choice.
• If we think of the expression in the given series as a function and like the idea of integrating it, then (assuming that this function is non-increasing) we can use the Integral test.
• We should definitely know the p-test. Sometimes one can determine convergence via geometric series or using the telescopic series trick.

Note that for many series one has the choice between the Ratio test and the Root test, then it is a matter of personal preference. If both tests are applicable, then the appropriate two limits yield the same answer.

Consider a series  ∑ a_k  of real numbers. We say that this series is alternating if the signs go + − + − + −... Formally, there exist positive numbers b_k such that a_k = (−1)^kb_k or a_k = (−1)^k+1b_k (depending whether we subtract every odd term or every even term), on both cases one has b_k = |a_k|. Such a series is the best behaved series among those whose signs change, in particular because there is a simple test.

If the sequence {b_k} is non-increasing and tends to 0, then the series  ∑ a_k  converges.

For an example, see Alternating series test in Convergence of general series in Theory - Testing convergence and this problem and this problem in Solved problems - Testing convergence.

Consider a series  ∑ a_k  of real numbers whose signs are not the same nor alternating, so the above two kinds of tests do not help. In that case probably the most useful approach is to try absolute convergence, namely apply the following fact.

If the series  ∑ |a_k|  converges, then also the series  ∑ a_k  converges.

The series with absolute values then falls into the category "series with non-negative terms", so all kinds of criteria are available to test its convergence. Note that the statement above is an implication. If the series with absolute values turns out to be divergent, then nothing follows about convergence of the given series.

There is one exception to this rule. If we tested the absolute convergence using the Root or the Ratio test (applied to |a_k|), then we have the following facts.
If ϱ < 1 (or λ < 1), then the series ∑ a_k converges.
If ϱ > 1 (or λ > 1), then the series ∑ a_k diverges.
(See Convergence of general series in Theory - Testing convergence.)

Sometimes (but rarely) one can show that the given series diverges by proving that its terms when taken as a sequence do not tend to 0 (see the necessary condition).

There are some tests that could help also with series whose signs are not alternating or the same, for instance Dirichlet's test, but these are rarely covered in typical calculus courses and thus they are beyond the scope of Math Tutor. Still, we decided to show some examples, see this problem and this problem in Solved Problems - Testing convergence.

Consider a series  ∑ a_k  of real numbers. We want to know whether it converges absolutely.

Definition says that it converges absolutely if the series  ∑ |a_k|  converges. Thus testing for absolute convergence of the given series is simple, we ask about the convergence of  ∑ |a_k| and here we can apply the tests for series with non-negative terms above.

For an example see for instance this problem and this problem in Solved Problems - Testing convergence.

Consider a series  ∑ a_k  of real numbers. To investigate its convergence means answering the following

Question: Does the series converge? If it does, then how?

In general, answering such a question means that we have to test two things:
a) Is the series convergent?
b) Is the series absolutely convergent?
We saw above how to answer these two questions.

Depending on the two answers, the overall answer to our classification question may be
  • "diverges" - if the answer to both a) and b) is "no"; note that in fact once we get divergent for a), we automatically get divergent for b), so in this case we need not ask both questions.
  • "converges conditionally" - if the answers are "yes" for a) and "no" for b).
  • "converges absolutely" - if both answers to a) and b) are "yes"; note that once we get "yes" for b), we automatically get "yes" for a); thus if we happen to start from the right end, we need not do the work for a).

For an example see this problem, this problem, this problem, and this problem in Solved Problems - Testing convergence.