Educator

Comparison Test

Main theorems:

Suppose and are series with positive terms. ( is given to you, and you create yourself.)

Comparison Test:

• If a_n < b_n for all n and converges, then converges.

• If a_n > b_n for all n and diverges, then diverges.

Limit Comparison Test: Suppose lim = some finite, positive (not 0) number.

• Then if converges then converges, and

• if diverges then diverges.

Hints and tips:

• Like the Integral Test, the Comparison Tests only work for series with positive terms. However, after you learn about absolute convergence later, you may be able to use them for series with some negative terms by taking their absolute value and seeing if they are absolutely convergent.

• The idea with these tests is that you are given a series . You come up with the series yourself. It should be similar in form to , but simpler to analyze.

• Often you will take to be a p-series or geometric series. Remember the differences between these two and the conditions under which each one converges or diverges.

• The Comparison Tests are two-way tests − you can use them to conclude that a series converges or that it diverges. However, it’s very important that the inequalities go the right way. If you have a series that is bigger than a known convergent series, the Comparison Test tells you nothing.

• And if you have a series that is smaller than a known divergent series, the Comparison Test tells you nothing.

• Because the inequalities must go the right way, it’s often useful to get some intuition about whether a series converges or diverges before setting up a comparison. It’s useful to remember the ranking of functions:

  

• When you have a complicated rational expression, focus on the biggest term in the numerator and the biggest term in the denominator.