Infinite Series Strategy Sheet

Know the Famous Series

Did I mention you should know the Famous Series?

Test for Divergence

Unless you immediately recognize a series (e.g. it's one of the Famous Series, or an obvious candidate for one of the Convergence Tests) always start with this one. Why?

At best, you could be done already!

At worst, you'll have a great idea how to proceed:

then you don't know what happens. Despair Not! Your work was not in vain.

• Ask yourself: How does go to zero?
• In the limit, does resemble terms in a famous series?
• Does the famous series have known convergence properties?
  • If so, the problem series almost certainly has similar convergence properties. Set up a comparison with the famous series. The Limit Comparison Test is usually a good bet here, since you've already been looking at a similar limit.
  • If not, go back and review Famous Series it's probably there.

Did I mention you should know the Famous Series?

Limit Comparison Test

If you don't find an easy match to a Famous Series, The Test for Divergence will almost always provide you with a Famous Series to use with the Limit Comparison Test. Set up the ratio between individual terms of the unknown series and the Famous Series and find the limit, L . If 0 < L < ∞ then the two series behave the same. If L is 0 or ∞ with any luck, your Famous Series "wins" the limit of the ratio in a useful way:

• Your unknown series converges if it is clearly smaller than a convergent Famous Series -- think about it.
• Your unknown series diverges if it is clearly larger than a divergent Famous Series -- think about it.

Did I mention you should know the Famous Series?

Convergence Tests

What are the various Convergence Tests?

Which one should I use?

You have a number of Convergence Tests available, and most series can be analyzed with more than one of them. The Convergence Tests page has guidelines for diagnosing when a test is likely to work on a particular series.

-- DickFurnas - 17 Nov 2008