Convergence tests do not belong in calculus

The power series chapter of any modern calculus textbook contains two topics:

(A) The power series for the elementary functions and their applications.

(B) Rigorous theory of convergence, including various “convergence tests.”

In fact, most courses teach no genuine applications of power series at all, yet go through tons of intricate problems on convergence tests. The student is left with the impression that all one ever does with power series is check whether they are convergent or divergent.

It is as if someone offered a course called “Cooking with Mushrooms” which, however, contained no actual cooking or eating or food of any kind. Instead it was concerned only with theoretical taxonomy and intricate technical methods for telling whether a given mushroom is edible or poisonous. Of course someone should study those things, and the rest of us will be grateful to benefit form their expertise. But such theories are not an end in itself. Telling which mushrooms are poisonous is only a meaningful problem if you are in fact going to cook and eat mushrooms. And in an introductory class it would of course be much more rewarding to learn to recognise and work with some common mushrooms than in going over theoretical points regarding the organic chemistry of obscure mushrooms that only grow in northern Madagascar.

So also in a calculus course. Learn first to appreciate power series. Get a taste for them, learn to work with them, and see what interesting compositions can be cooked up using this ingredient. Of course everyone needs to be aware that some series are “poisonous,” but some basic rules of thumb are enough to avoid the associated dangers. Don’t sit in a sanitised room obsessing over potential dangers like a hypochondriac. Put some sturdy boots on and go into the wilderness and explore.

This is also how it worked historically. As history shows, (A) and (B) are two independent topics. (A) was worked out in great detail already in the 17th century, while (B) only emerged in the early 19th, some 150 years later. (B) was developed not because it is needed for (A)---indeed history proves that it is not---but in response to much more refined technicalities.

(B) serves no credible purpose in a calculus course. It is included solely for the sake of rigour machismo. It would be bad enough if this was the end of the problems with these courses, but it is worse still. For, as a rule, calculus teachers either do not realise or not not care to admit that rigour machismo is the only rationale for this material. Instead they pretend that it is necessary to avoid various pitfalls and paradoxes that would supposedly plague anyone who dared take a step into the jungle of power series without being armed with ratio, root, and integral tests. Such fear-mongering and quasi-justification is nothing but deceit and indoctrination. It has no basis in fact, as history makes abundantly clear.

The right way, and the historically accurate way, to teach power series is to start with their uses. Brave and daring use of power series is exhilarating and rewarding and quickly leads to many fascinating results. True, it soon becomes evident that such methods are not always valid, and that too careless trust in blind manipulations of power series can quite easily lead to errors. But such matters are easily handled by some basic common sense and the most rudimentary distinction between convergence and divergence.

The wrong way, and alas the common way, to teach power series is to say very little about their actual uses---that is, the reasons for why people studied them for hundreds of years before there was any such thing as convergence theory---and instead focus only on how to tell whether a given series is convergent or divergent. This is doubly stupid: firstly because it vastly overstates the importance of being able to decide convergence questions for arbitrary series, and secondly because it is concerned virtually exclusively with series made up for the sole purpose of being suitably difficult to apply convergence tests to, instead of studying the many fascinating situations in which power series emerge naturally in a way that actually makes sense and therefore can be introduced in an intellectually responsible and honest, rather than dogmatic and dictatorial, manner.

I am by no means opposed to convergence theory. It is a lovely and fascinating subject. But it should be taught as what it is. And calculus it is not.