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?