I have a general question when it comes to deciding if an infinite series is convergent or divergent. The tests im familiar with are ; Ratio test, Direct comparison test, Limit comparison test, Root test and the Integral test.
My question is if there is any way to tell what test is appropriate to start with just by looking at the series. At the moment I usually follow my own pattern and systematically try different tests. What I do is:
- Divergence test to see if the series is divergent. If its unsuccessful
- Ratio test, if unsuccessful
- Limit comparison test
Which has been working fine, my concern is that it may be very time consuming if the first tests are unsuccessful, and might not be very efficient during exams.
So basically: Is there a way to determine what tests are appropriate by just looking at the series? ( I have found nothing like this in my textbook, all the examples simply jump straight into the "correct" test).