In our previous lesson, Geometric Series, we learned how the common ratio enabled us to determine convergence/divergence of an infinite series, and also how to find the sum of a convergent series.
But sometimes it we are given a series where we can simply compare one function to another and determine convergence.
The idea behind the comparison test is to graph our function and then compare it with a graph of a known expression that is similar to the one we are asked to evaluate. Always making sure that our chosen comparison function is above, or on top, whereas our given function is underneath, or on the bottom.
Comparison Theorem for Improper Integrals
Now we use the comparison theorem to determine convergence or divergence.
- If the area under the graph of our given function (the one on the bottom) is infinite then the area under the comparison function (the one on top) will also yield an infinite area. Therefore, both functions diverge.
- If we can show that the area under the comparison function (curve on top) is finite, then the area under our given function (curve on the bottom) is too; thus convergent.
But what happens if you don’t know what the graph of the function looks like? Or you don’t know what to compare with?
Don’t worry! The ever useful Limit Comparison Test will save the day!
Definition of the Limit Comparison
The Limit Comparison Test is easy to use, and can solve any problem the Direct Comparison Tests will solve. Yay!
Therefore, out of the two comparison tests, the Limit Comparison Test is the most important and helpful.
The comparison test determines converges or diverges by comparing it to a known series.
What is important to note is that it is critical that we compare our given series with a known, “easy” series.
In this lesson we will see countless examples of how to create a series that is necessary to compare, and be able to determine convergence/divergence quickly and efficiently.
Limit Comparison Test Video
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