[Math] Strategy to recognize and solve sequence and series problems

calculussequences-and-seriessoft-question

I've been reading my Stewart Calculus book and I honestly find most of the coverage of sequences and series easy to grasp (excluding power series, Taylor and Maclauren since we haven't covered those just yet).

However, when the book guides me to a test to use to solve a problem I don't have an issue, but if I'm given a bunch of problems and told to choose a test to use I honestly have no idea where to begin.

How can I intuitively know how to attack a given problem? I realize the thought process is quite similar to integration, but I still feel like integration is much easier than determining convergence and divergence of a sequence/series.

Best Answer

Consider $$L = \sum_{n=1}^\infty \frac{(-1)^n}{n^2}$$ This qualifies for both, comparison and Leibnitz:
Leibniz is applicable because $\frac1{n^2}$ is a non-increasing, positive null-sequence in $n$
And comparison yields $$|L| \leq \sum_{n=1}^\infty \left| \frac{(-1)^n}{n^2} \right| = \sum_{n=1}^\infty \frac1{n^2} = \frac{\pi^2}6$$

Worth noting here is, that using comparison as an Idea, we have even proved absolute convergence, which is stronger than plain convergence of Leibniz, but was a little bit more work.

A few general steps are:

  1. Is $(a_n)_n$ even a null-sequence?
  2. Can I prove absolute summability (then I won't have to worry about signs)? This allows for lazy estimates like $\sin, \cos \le 1$ but cannot prove divergence
  3. Is Leibnitz applicable (if alternating)?
  4. Do the terms look like telescoping? Something like $a_n = f(n) - f(n+a)$
  5. Does the series look like another series, which I could compare it to?
    Most notably $\sum\limits_{n=a}^\infty \frac1{n^\alpha}$, converging iff $\Re\alpha > 1$
  6. Are the coefficients "nice" in a way that I can use integral comparison?
  7. ....
Related Question