Strategies for infinite series: which test do I use?

College infinite infinite series mathematics
By Rhonda

Because there are so many tests for determining the convergence or divergence of an infinite series, students often feel overwhelmed when faced with a series and have to decide whether it converges (C), diverges (D), converges absolutely (CA), or converges conditionally (CC). Here are some strategies that might bring a little order to the chaos! 

Disclaimer: This blog assumes you have already seen the tests but just need some help organizing your thoughts.

An infinite series looks like this:

Screen Shot 2025-01-06 at 9.58.24 AM

Here are the tests we have available for an infinite series:

  • nth Term Test (for divergence only!) 
  • Geometric Series 
  • Integral Test 
  • p-Series 
  • Comparison Test/Limit Comparison Test (LCT) • Alternating Series Test (aka AST) 
  • Ratio Test 
  • Root Test

 

Strategy 1: Often overlooked, the nth Term Test is frequently a last resort when nothing else is working.

To save yourself time and aggravation, check the following first:

Screen Shot 2025-01-06 at 10.03.29 AM

If not, the series diverges, and you are done! So keep an eye out for terms that look like they don’t get small.

Screen Shot 2025-01-06 at 10.04.37 AM

Strategy 2: Does the series look like a geometric series (An = rn for some real number r)?

If so, use the criteria for determining whether a geometric series converges or diverges.

Screen Shot 2025-01-06 at 10.06.48 AM

Strategy 3: Is the series a p-series? Make sure you know the difference between a geometric series and a p-series!

Screen Shot 2025-01-06 at 10.07.26 AM

Strategy 4: If the terms are quotients of powers of n, check the highest powers of n in the numerator and denominator, and use the Limit Comparison or Comparison Test.

DO NOT use the Ratio Test in this case! You will ALWAYS get a limit of 1, in which case the Ratio Test fails.

Screen Shot 2025-01-06 at 10.09.24 AM

Strategy 5: If the terms of the series involve ln n or en, try the Integral Test.

Screen Shot 2025-01-06 at 10.10.53 AM

Strategy 6: Do the terms of the series involve factorials? If yes, the Ratio Test is a good bet.

Screen Shot 2025-01-06 at 10.11.25 AM

Strategy 7: If the terms of the series involve expressions raised to the nth power, try the Root Test, unless the base is a constant, in which case the series is geometric.

Screen Shot 2025-01-06 at 10.12.17 AM

Strategy 8: If the terms alternate in sign, first see if the series converges absolutely (CA), using one of the tests above. If it does not converge absolutely, try the Alternating Series Test.

We're done and the series diverges if the following is true:

Screen Shot 2025-01-06 at 10.13.59 AM

If the limit is zero, and the terms decrease, then the series converges conditionally (CC).

Screen Shot 2025-01-06 at 10.14.49 AM

Are we guaranteed that these strategies will always work?

Not completely. There are some tricky series that might escape our grasp, despite these tips; nevertheless, this post should take you a long way in improving your series game!

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