Wednesday 8 November class


• Theorems about convergence/divergence of series

So far we have the following:

• The integral convergence test (from last time):

• (Direct) Comparison test: this is very similar to the direct comparison test for improper integrals where the upper bound of integration is infinite.

• Limit Comparison test: this is also very similar to the limit comparison test for improper integrals where the upper bound of integration is infinite.

Strategy to choose a method: I would usually try them in the order given above. If it is easy to do integral test, use that. (Maning, you can easily figure out whether the integral converges or diverges.) If that is not easy, see if there is a direct comparison with some series you know about (a geometric or p-series, especially). If the direct comparison does not work because the inequality goes the wrong way, it is very likely that you can use limit comparison.

I worked these examples using the integral test:

The harmonic series \sum_{n=1}^{\infty}\frac{1}{n} diverges, because we can compare to the integral \int_{1}^{\infty} \frac{1}{x}\textrm{d}x and we show (or remember) that this diverges.

Note: the harmonic series is just a p-series with p=1. Similarly, we can show that any p-series will converge if p>1 and diverge if p\le 1, by comparing to the appropriate improper integral.


Show that \sum_{n=0}^{\infty} ne^{-n^{2}} converges: this example will show you a couple of important considerations in using the integral convergence test which you may be tempted to skip over, so study it carefully.

Unfortunately, the LaTeX rendering in WordPress is being very glitchy so I cannot just type this example in this post. I have made it in pdf form for you:


You can also find a discussion of this example in Paul’s Online Math Notes. (example 2 on that page)

Here are some links to relevant parts of Paul’s notes:

Integral test notes

Integral test practice problems

Comparison test and Limit comparison test notes

Comparison test and Limit comparison test practice problems

It’s very good to use those practice problems to test yourself, as you can see his solutions right away when you finish. Testing yourself is one of the best ways to learn.


• Review the examples discussed in class for all 3 of these tests. You can (and should!) also look at Paul’s Online Math Notes for even more examples and practice problems.

• Do the WeBWorK assignments on Intro to Series 2 (which has been extended) and Comparison Tests. Another assignment was also posted but you do not need to work on it yet.

• Don’t forget that Test 3 is scheduled for next Wednesday. The review materials will be in a separate post.



Don’t forget, if you get stuck on a problem, you can post a question on Piazza. Make sure to give your question a good subject line and tell us the problem itself – we need this information in order to answer your question. And please only put one problem per posted question!