What is the "candidates test" for convergence?

This is a serious answer, I have never heard of any "candidates test" and I taught calculus for years at various levels, and at various schools including teaching-oriented universities and research oriented universities. I have worked with, and talked about teaching calculus with, people who have taught it for decades. I can almost guarantee you that none of them have ever heard of any test called the "candidates test", or I almost certainly would have heard of it. The name is sufficiently vague that I can't imagine what it might be referring to. The inescapable conclusion of this is that the term "candidates test" was invented by that teacher, and that what it refers to cannot be determined without asking that teacher.

For what it's worth, even if you do talk to that teacher and learn what that test is, please do not call it that in front of your students. It is hard enough to teach math without having to deal with people inventing their own terminology for things and passing it on to their students without explaining that they are simply making up a name for something.

Don't get me wrong--- I think it's useful to invent names for specific things that are found useful in specific classes. And I don't think it's reasonable, or even desirable, for there to be some central authority to impose a naming convention on the entire teaching community. But when people do invent names for stuff, as is often necessary, it is irresponsible not to communicate to your students and fellow teachers that you are simply inventing a name for something that will be used so often in your class that it needs a name. Genereally, the less we cause our students to think "gee, I'm completely unfamiliar with that and have never heard of it before, it must be some universal math thing", the better off we are. Students need to be told when things are part of established terminology and when things are just made up