# Proving convergence or divergence of series: tips and recommendations

This is a follow up of my question on MSE. Which tips and recommendations would you give students who want to investigate series about convergence or divergence?

So far we have collected:

Which recommondations would you add to the above list?

• Maybe emphasize more the importance of pattern-recognition. You mentioned p-series, but a student should also be able to quickly recognize geometric and alternating series (of the type easily shown to be convergent). Furthermore, students should be able to recognize when something is asymptotically equivalent to one of these known series (this is often formalized by the limit comparison test). For example, they should be able to look at a sum with terms of the form 1/(2n+1) and be able to instantly recognize it as being essentially the same thing as a harmonic series. – John Coleman Jul 28 '16 at 11:32
• Beyond these excellent suggestions already made, it's also important to have some sense of relative growth. For example, polynomials always lose to exponentials, but exponentials can't survive a factorial... – James S. Cook Aug 18 '16 at 1:50

The heuristics I emphasize in my AP Calculus class are as follows:

(1) Check the nth term test before you do anything else. (As noted in your list)

(2) Check to make sure you don't have a straightforward p-series, alternating series, or geometric series before proceeding with the other alternatives.

(3) A limit comparison test to an appropriate p-series almost always works for quotients involving polynomials and similar expressions (by similar, I mean expressions which have fractional powers instead of just non-negative integers, etc.).

(4) The ratio test almost always works for series involving exponential expressions and factorials. (Yes, the root test is also excellent for exponentials, but I tend to emphasize the ratio test since it's more versatile. I would not advise having students jump to the root test in most other situations.)

There are a handful of oddballs for which these heuristics don't work, but these cover a vast majority of the commonly encountered problems.