We have an entire tool belt full of convergence tests to determine if series converge or diverge. The tests are as diverse as tools on a tool belt, too. The divergence test is a drill to bore holes into a series we suspect diverges, and the comparison test is a tape measure for us to compare to other series. We can even use the relationships of absolute and conditional convergence like a hacksaw to cut a series in half using the absolute value to check for absolute convergence. Tim "The Tool Man" Taylor would approve.
Now that we have all of the these fancy tools hanging around our waist, we need to now which tools to use when. It takes a bit of a mental jump to go from problems that say
"use the ratio test to determine if this series converges"
to problems that say
"determine if this series converges"
without giving any hints about which test(s) you should be using.
We presented the tests more-or-less in the order you should try them.
- Try the divergence test. It's easy to check if the terms of a series converge to 0. If they don't, you know the series diverges and you don't have to do any more work. Almost as easy, is the series a geometric series? If so, is |r| < 1?
- Is the series an alternating series? If so, try the AST.
- Try the integral or ratio test, depending on what the terms look like. It probably won't make sense to try both the integral test and the ratio test.
(a) If it looks like the ratio will simplify nicely, try the ratio test. Exponents and factorials are good indicators.
(b) If it looks like the terms are given by an integrable function, try the integral test.
- The comparison test is the only one left, so try that. You may also need to use some other test to show the convergence/divergence of the series you're using for comparison.
- One other idea to have in the back of your head: if the series converges absolutely, then the series converges. If the divergence test and AST don't help, you might want to ask yourself if showing absolute convergence is the easiest way out.
Just like woodworking or machining a widget, the best way to get better is to practice. Learning to use the right tools can be frustrating in any situation. We should keep that in mind, being grateful that these aren't real power tools. We can't cut a thumb off using the alternating series test.
There are multiple right ways to solve most of these problems. If you use a way we didn't mention, check with someone else to see if you found another correct way.