The first test is the comparison test. Think of a converging series as one which is “small”, that is, the terms go to zero so rapidly that the infinite sum converges to a finite real number. If another series is “smaller” than a converging series then it too should converge. converges if the sequence of partial sums converges and diverges otherwise. For a particular series, one or more of the common convergence tests may be most convenient to apply. Lectures 11 - Infinite Series, Convergence tests, Leibniz's theorem. Series: Let (an) be a sequence of real numbers. Then an expression of the form a1 +a2.


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The divergence test is the first test of many tests that we will be looking at over the course of the next several sections.

Convergence tests

You will need to keep infinite series convergence tests of all these tests, the conditions under which they can be used and their conclusions all in one place so you can quickly refer back to them as you need to. Furthermore, these series will have the following sums or values. At this point just remember that a sum of convergent series is convergent and multiplying a convergent series by a number will not change its convergence.

We need to infinite series convergence tests a little careful with these facts when it comes to divergent series.

List of Series Tests

Now, since the main topic of this section is the convergence of a series we should mention a stronger type of convergence. Absolute convergence is stronger than convergence in the sense that a series that is absolutely convergent will also be convergent, but a series that infinite series convergence tests convergent may or may not be absolutely convergent.

When we finally have the tools in hand to discuss this topic in more detail we will revisit it.

The idea is mentioned here only because we were already discussing convergence in this section and it ties into the last topic that we want to discuss in this section.

First, we need to introduce the idea of a rearrangement. A rearrangement of a series is exactly what it might sound like, it is the same series with the infinite series convergence tests rearranged into a different order.

Series convergence & estimation | Khan Academy

For example, consider the following infinite series. Here is an example of this. The values however are definitely different despite the fact that the terms are the infinite series convergence tests.

Here is a nice set of facts that govern this idea of when a rearrangement will lead to a different value of a series.


infinite series convergence tests Once this has been done you can identify the test that you feel will be the easiest for you to use.

With that said here is the set of guidelines for determining the convergence of a series. If so, use the Divergence Test.

Note that you should only do the Divergence Test if a quick glance suggests that the series terms may not converge to zero in the limit. Remember as well that often some algebraic manipulation is required to get a geometric series into the correct form.

If it converges determine its value. So, as we saw in this example we had to know a fairly obscure formula in order to determine the convergence of this series. In general finding a formula for the general term in the sequence of partial sums is a very difficult process.

  • Series convergence & estimation | Khan Academy
  • Convergence tests
  • The Common Series Tests
  • Convergence tests for infinite series

We will continue with a few more examples however, since this is technically how we determine convergence and the value of a series.

Example 2 Determine if the following series converges or diverges.

List of Series Tests

If it converges determine its sum. Therefore, the series also diverges.


Example 4 Determine if the following series converges or diverges. Two of the series converged and two diverged. This will always be true for convergent series and leads to the following theorem.

This theorem gives us a requirement for infinite series convergence tests but not a guarantee of convergence.

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