TITLE: How can I determine which series comparison test to use?
QUESTION [8 upvotes]: In my textbook, there is a section of questions that's instructions reads "Test for convergence or divergence, using each one of the following tests once," and the test choices it gives me are

nth-Term Test
p-Series Test
Integral Test
Limit Comparison Test
Geometric Series Test
Telescoping Series Test
Direct Comparison Test

Now, I was able to get most of the problems in this section right on the first try, but when I compared my answers to those in the solutions manual, I found that often the way I chose was not the same as theirs. Sometimes, my solution was simpler, but most of the time, the method they chose was quicker. For one of the problems, I did the limit comparison test and I ended up with a $b_{n}$ that was one, so I was really just doing some convoluted mix between the Limit Comparison Test and the nth-Term Test.
How can I develop skills to help me to decide which method to choose? Is there some "trick" to deciding which method will be the simplest?

REPLY [4 votes]: The p-series and geometric series tests are for specific types of sequences, and it is clear when you can apply them.
Use the integral test for positive, decreasing functions or negative, increasing functions only (do not forget this condition).
Telescoping series always look like $\sum f(x+1)-f(x)$, so like the other series, they are for a particular type of series but watch out for the series $\sum \frac{1}{n(n+1)}$ and similar series that can be made into a telescoping series using partial fractions.
The best thing to do is to simply practice exercises until you have mastered their usage.