If an = f(n) for a nice-to-integrate function f(x), that's a good hint to try the Integral Test. (Don't forget to check to see if f(x) is positive, continuous, and decreasing on the domain you need.)
Try the Ratio or Root Test. Integral-looking terms? Try the Integral Test. Need comparison? Try the Comparison or Limit Comparison Test. Not sure? Use Ratio or Root Test as general go-tos! Is it a geometric series? Is it an alternating series? Try this if your series looks similar to a known p-series or geometric series. Good for nth powers..
It outlines several common tests for convergence, including the p-series test, geometric series test, alternating series test, telescoping series test, Taylor series test, comparison test, limit comparison test, integral test, and ratio test.
This worksheet contains a flowchart to help you choose a convergencetest to use. When given a series, you can follow the chart to determine which test to use to analyze that series.
n 1 np converges if and only ifp > 1 Geometric series: Ifjrj< 1 then X1 n=0 arn= a 1 r otherwise, that series diverges. All that matters is what happens on a tail. For example, ifa nis only decreasing after N then you can write: X1 n=1 a n= XN n=1
Now that we’ve got all of our tests out of the way it’s time to think about organizing all of them into a general set of guidelines to help us determine the convergence of a series.
