I taught this subject only once (two years ago). In that occasion, I wrote an one page summary for my students in the following order.
I think this is one of those places where teaching a detailed strategy is a form of "teaching to the test" that is counterproductive for the students' intellectual development.
It's important to emphasize that all the convergence tests have preconditions that must be satisfied before one can use the rule, and one should not use a test without ...
First, let's ignore geometric series and $p$-series because those are standard examples.
For infinite series with positive terms, if you really understand how sequences grow then almost all examples that occur in basic courses can be handled by the limit comparison test unless the sequence has factorials in it, in which case you use the ratio test.
The root ...
I don't teach this course, and I think the desire to have an algorithm this detailed may be an example of the kind of thing that people will start to feel because they're teaching the class but that people in the real world don't actually use. Students want to be told that everything is rule-based, because it makes them feel safer.
When I encounter this kind ...