Testing Convergence of Series

Here we will address the problem of testing convergence of series. If you wish to simultaneously follow another text on testing convergence in a separate large window, click here for Methods Survey and here for Solved Problems.

Testing convergence of a series that is not in some way special is very difficult and there are no reliable methods for it. Probably the most general statement we offer is the necessary condition for convergence, see the appropriate theorem in Testing convergence - Convergence of general series. It stipulates that a series diverges if the individual terms of it do not tend to zero as a sequence. While it formally does apply to all series, in most cases it is useless as it offers no information on convergence of series whose terms do go to zero. As can be expected, these are exactly the series we are usually interested in.

One reason why a general test of convergence is hard to come by is the wide freedom of behavior that series can exhibit. In particular, a series can diverge in many different ways and it is hard to capture them all in one test. Thus the natural thing is to look at some special series. The obvious choice is motivated by the appropriate theorem in section Basic properties in Theory - Introduction to series. It essentially says that if in a series all terms have the same sign (zeroes are also allowed), then such a series has only two options, either it converges or it sums up to infinity (or negative infinity, depending on the sign of its terms). It turns out that in this simpler setting (convergence or infinity) it is much easier to distinguish between the two cases, because then it is just a question of how large the terms of the given series are. It is enough to consider series whose terms are all non-negative (positive or zero), since in a series with all terms non-positive we can factor the common minus out of the series and we get a series with non-negative terms.

Below we will show some very powerful tests of convergence for series with non-negative terms. While this condition may seem quite restrictive, it is not so bad. Many series fall into this category. Moreover, we often investigate absolute convergence where we apply absolute value to individual terms of the given series, therefore all the tests for non-negative series become available. This is actually one of the strategies recommended in the last section here, where we show some tips on how to approach series with changing signs.