Which test for convergence is the best for which series?

In IB Calculus Option you are introduced to number of different tests. They are Divergence Test, Comparison Test (simple and limit forms), Absolute Convergence Test, Alternating Series Test, Ratio Test and Integral Test. I think it is always good to start with Divergence Test, because checking if the limit of the terms of the sequence is 0 is usually really quick and it can save you time if your series is divergent. If your series has both positive and negative numbers check if it is an alternating series, if yes then use the associated test. If it has both positive and negative terms but is not alternating, then maybe try simplifying situation with Absolute Convergence Test. If series looks similar to something that you know, then Comparison test usually does it job. It is also easy to apply. If it is too messy with using inequalities use its limit form.
Ratio Test is a powerfull tool, but it is proved by comparing it to a geometric series. Hence it works well if your series looks similar to a geometric one. It may have some extra expressions that maybe simplify easily while calculating the limit of the ratio of consequtive terms. It works well with powers and factorials. Not neccesairly good for other types of series.
The most powerful tool introduced in the course is the Integral Test. It often provides us with insight into convergence of series that was not avalible with previous tests. It is useful when you can integrate the functions that generate your series. If not, then it is not worth to look at it. You will often see that sometimes combining two tests is necessary. Especially when using Absolute Convergence Test, since it generates you a new series of positive terms, which convergence you have to determine using other methods.