Proving convergence or divergence of series: tips and recommendations

Question

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:

• It is good to start with the term test.
• For Series of the form $\sum_{k=1}^\infty a_k$ the root test is often useful.
• $\sum_{k=1}^\infty \frac{1}{k^p}$ converges for $p>1$ and diverges for $p\le1$.
• Krummer's test is also useful.

Which recommondations would you add to the above list?

Answer 1

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.