Just Go with the Flow: A New Strategy for Teaching Convergence Tests
Series.
The one little word that conjures up fear in the hearts of many Calculus BC students (perhaps in also a few new BC teachers?). And those convergence tests! How will we ever keep them straight or, more importantly, know which one to use when? This was also the plight of my students until I decided to give them a visual of “the big picture” in the form of a flow chart. Although there is still a struggle to learn the tests, my students tend to embrace this part of Unit 10 a bit more willingly. Maybe they have learned to go with the flow?
Here is how the flow chart works:
1. Is this series in my toolbox?
The idea of having some series that you “just know” is a powerful tool for students new to the convergence game and I try to emphasize the importance of recognizing these series quickly and determining their convergence without any further work. The three series in my toolbox are:
The Harmonic Series - which diverges
TheGeometric Series - which converges if |r| < 1 and 3
The P-Series - which converges if p > 1. (Note: the key difference between geometric series and p-series is where the variable “n” is located…an important point, I think!).
You may want to add other series to your toolbox (such as the telescoping series), but as far as the CED is concerned, these three should suffice.
2. The nth term test
If the series isn’t one we know, what are the terms doing? Are they headed toward zero? This test is so often forgotten by students, but it can lead them to a conclusion of divergence quickly if that limit is not zero!
3. Is this similar to a series in my toolbox?
There are many examples of series that look “almost harmonic” or “almost geometric” but vary due to a constant added somewhere. If this is the case, the Direct Comparison Test or Limit Comparison Test will quickly resolve the issue!
4. Nothing works! Now what??
Of course there will be plenty of situations where the series is not recognizable and the terms do not approach zero. Then what do we do? Never fear…there are three other options available depending on the series that is presented. Is it an alternating series? Use the Alternating Series Test! Are there factorials and/or exponentials in the formula? Try the Ratio Test! Could you integrate this expression, perhaps using substitution? Then the Integral Test is your best bet!
There will always be series that seem to defy all tests, but with this flow chart, at least students can feel a sense of agency and ownership over the process. More importantly, the ability to make the decision of which test is the best choice becomes a learnable skill rather than a shot in the dark from an overwhelming list of options.
ONE NOTE OF CAUTION: Although this flow chart does lead students to the correct test to determine convergence, it still is important to emphasize that when a test is applied, the requirements for that test must be examined and stated, just as we would if we were using the Mean Value Theorem or L’Hospital’s Rule. The purpose of this flow chart, then, is not to be a comprehensive “how to” manual but rather a map to guide students in the right direction. Once they have selected a test, the rest is up to them!