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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Everything was explained so well. It was easy to follow along and understand.
Its been super helpful and goes in depth and explains it so well even a 5th grader could understand it.
I appreciate that all work is shown so that I am not confused on how to get from one step to another.
I haven't used it much but so far it's been a great experience
The material for calculus has been extremely thoroughly and helpful. She explains it better than my instructor! I've even been able to refresh on concepts from the past to fill in holes from other years and semesters. I finally feel like I can actually take my Calculus class without failing!
I am 47 years old and a freshman in college. I started studying with Jenn at calcworkshop a few months before my first semester and during. I am proud to say I achieved a 4.0 GPA and a spot on the Dean's list. Thank you so much! Great learning tool here!!!
I could not recommend calcworkshop enough; the style and format Jen instructs in is amongst the best I have seen, and I would not have made it through my mathematical journey without her. I recommend this website to anyone who ever confides in me that they are having troubles in any math class, not just calculus. Forever grateful!
The videos were extremely helpful! Thanks so much for explaining everything in detail!
I started w/Jenn because I love math and have a reading handicap. I am very happy to start from the beginning again with her help and guidance. I can tell Jenn loves math as much as I do and she is willing to teach the beginning with as much zeal as the advanced.
My experience has been great so far! I purchased this because I got frustrated searching YouTube for how to information. It is affordable, great supplemental information to my course!