Strategies for infinite series: which test do I use?

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:

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

• n^th 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 n^th Term Test is frequently a last resort when nothing else is working.

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

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

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

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

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

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.

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

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

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.

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:

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

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!