In this video se are going to talk about the Comparison Test. The way this works is that we will be given a series that we are going to call an, and we create our own series that we will call bn. Then we will try to compare these two series to each other. So, first, we are going to talk about how to determine when the given series converges and when it diverges using the Comparison Test. We will see some examples that illustrate what happens when the test goes the right way and the wrong way. Then we are going to talk about second test called the Limit Comparison Test and see how it works.

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Last reply by: Dr. William Murray Thu Aug 4, 2016 6:02 PM

Post by Peter Keon July 30, 2016

For example 3, why can't we go straight to testing if the exponent is bigger than or less than 1 instead of wasting time doing the limit comparison test?

1 answer

Last reply by: Dr. William Murray Thu Apr 24, 2014 6:18 PM

Post by Brandyn Albrechton April 22, 2014

Quick question. taking an/bn is just to figure out if the series are similar or whatever in terms of divergence, but for example 1 just to double check I took (1/n+1)/1/n and got n/(n+1) which converges to 1, which is a positive real number, which means an and bn should act similarly right? But they don't because an is smaller than bn, so does the an/bn approach only work for more confusing questions like example 3, and not for all questions in general?

3 answers

Last reply by: Dr. William Murray Tue Dec 17, 2013 9:09 PM

Post by Xenia Jeantyon November 30, 2013

lecture example 3 why is sqrt n/n same as 1/sqrt n? Can you please explain how did we get there?

1 answer

Last reply by: Dr. William Murray Tue Aug 13, 2013 5:24 PM

Post by charles danielon July 24, 2013

lecture example 3 in the denominator n/n^2 should be equal to zero ..right?..why is it equal to one

1 answer

Last reply by: Dr. William Murray Tue Aug 13, 2013 5:21 PM

Post by charles danielon July 24, 2013

lecture example 3 .. how is the bn of root(2n+17)/n become root n/n

1 answer

Last reply by: Dr. William Murray Tue Apr 16, 2013 8:27 PM

Post by Alena Schwartsmanon April 14, 2013

Thank you Dr. Murray, concise and to the point. I like how you repeat the test's requirements with every example; easier to memorize the tests this way. Overall, I love how you explain things! Thank you again!

1 answer

Last reply by: Dr. William Murray Wed Apr 3, 2013 6:43 PM

Post by Rohail Tariqon April 1, 2013

cool

1 answer

Last reply by: Dr. William Murray Wed Apr 3, 2013 6:42 PM

Post by Ahmad Alshammarion September 13, 2011

why you put .. 3^(n-2) !!

1 answer

Last reply by: Dr. William Murray Wed Apr 3, 2013 6:34 PM

Post by Srinivasa Rao RAVELLAon April 14, 2011

how would you should tan(1/x) is convergent with the limit comparison test

1 answer

Last reply by: Dr. William Murray Wed Apr 3, 2013 6:26 PM

Post by Romin Abdolahzadion April 4, 2011

Regarding the limit comparison test: If the limit is equal to 0, then if Bn converges then An also converges!

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.

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.

Comparison Test

Does the series ∑_{}^{} [1/(2 + n^{4})] converge?

Determine an appropriate u_{n} for the Comparison Test

Consider u_{n} = [1/(n^{4})]

Consider n = 2, then a_{n} ≤ u_{n}

∑_{}^{} [1/(n^{4})] p = 4 > 1, thus the series u_{n} converges. Since u_{n} converges, then a_{n} converges

Does the series ∑_{}^{} [1/(n^{2} − 50)] converge?

Determine an appropriate u_{n} for the Comparison Test

Consider u_{n} = [1/(n^{2})]

Consider n = 3, then a_{n} ≤ u_{n}

∑_{}^{} [1/(n^{2})] p = 2 > 1, thus the series u_{n} converges Since u_{n} converges, then a_{n} converges

Does the series ∑_{}^{} [1/(√{n + 6})] converge?

Determine an appropriate u_{n} for the Comparison Test

Consider u_{n} = [1/(n^{1/2})]

Consider n = 3, then a_{n}u_{n}

Determine ∑_{}^{} u_{n} diverges with P - series properties

∑_{}^{} [1/(n^{1/2})]

p = [1/2] ≤ 1, thus the series u_{n} diverges

Since u_{n} diverges, then a_{n} diverges

Does the series 1 + [1/2] + [1/(√7 )] + ... + [1/(√{3n − 2} )] + ... converge?

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.

Answer

Comparison Test

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

The book is filled with examples and detailed explanations. Capstone exercises are also included for each section and synthesize the main concepts into a single example.

1 answer

Last reply by: Dr. William Murray

Thu Aug 4, 2016 6:02 PM

Post by Peter Ke on July 30, 2016

For example 3, why can't we go straight to testing if the exponent is bigger than or less than 1 instead of wasting time doing the limit comparison test?

1 answer

Last reply by: Dr. William Murray

Thu Apr 24, 2014 6:18 PM

Post by Brandyn Albrecht on April 22, 2014

Quick question. taking an/bn is just to figure out if the series are similar or whatever in terms of divergence, but for example 1 just to double check I took (1/n+1)/1/n and got n/(n+1) which converges to 1, which is a positive real number, which means an and bn should act similarly right? But they don't because an is smaller than bn, so does the an/bn approach only work for more confusing questions like example 3, and not for all questions in general?

3 answers

Last reply by: Dr. William Murray

Tue Dec 17, 2013 9:09 PM

Post by Xenia Jeanty on November 30, 2013

lecture example 3 why is sqrt n/n same as 1/sqrt n? Can you please explain how did we get there?

1 answer

Last reply by: Dr. William Murray

Tue Aug 13, 2013 5:24 PM

Post by charles daniel on July 24, 2013

lecture example 3 in the denominator n/n^2 should be equal to zero ..right?..why is it equal to one

1 answer

Last reply by: Dr. William Murray

Tue Aug 13, 2013 5:21 PM

Post by charles daniel on July 24, 2013

lecture example 3 .. how is the bn of root(2n+17)/n become root n/n

1 answer

Last reply by: Dr. William Murray

Tue Apr 16, 2013 8:27 PM

Post by Alena Schwartsman on April 14, 2013

Thank you Dr. Murray, concise and to the point. I like how you repeat the test's requirements with every example; easier to memorize the tests this way. Overall, I love how you explain things! Thank you again!

1 answer

Last reply by: Dr. William Murray

Wed Apr 3, 2013 6:43 PM

Post by Rohail Tariq on April 1, 2013

cool

1 answer

Last reply by: Dr. William Murray

Wed Apr 3, 2013 6:42 PM

Post by Ahmad Alshammari on September 13, 2011

why you put .. 3^(n-2) !!

1 answer

Last reply by: Dr. William Murray

Wed Apr 3, 2013 6:34 PM

Post by Srinivasa Rao RAVELLA on April 14, 2011

how would you should tan(1/x) is convergent with the limit comparison test

1 answer

Last reply by: Dr. William Murray

Wed Apr 3, 2013 6:26 PM

Post by Romin Abdolahzadi on April 4, 2011

Regarding the limit comparison test: If the limit is equal to 0, then if Bn converges then An also converges!