Original subtitles

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In this video we're talking about how to use the limit comparison test to say whether a series converges

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or diverges.

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And in this particular problem we've been given the series from equals 1 to infinity of end squared

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divided by quantity and cute plus two.

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Now as a reminder the limit comparison test relies on a couple of conditions.

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First of all both the original series A7 and the comparison series be seben have to be greater than

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zero.

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So all the terms of the series have to be positive and in addition to that the limit as and goes to

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infinity of a sideband divided by B-7.

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So we're literally going to take the original series and divide it by the comparison series.

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If that limit the value of that limit is greater than zero then we can draw conclusions.

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We can say if the comparison series B-7 converges then the original series a seben also converges.

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Or if the comparison series B-7 diverges than the original series A7 also diverges.

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So we're going to first do is make sure that our original series and our comparison series meet these

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conditions before we go on to drawing any conclusion.

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So what will be our comparison series.

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Well what we usually want to do if we have a rational function or a fraction is we want to take the

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term from the numerator and denominator that's going to have the greatest effect on the numerator and

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denominator or the term with the greatest magnitude.

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So in this case there's only one term in the numerator it's end squared.

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So we're going to go ahead and take and squared for the new numerator and then for the denominator and

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cubed has a greater magnitude or greater degree than just the constant 2.

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So we're going to take and cubed only for the denominator.

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This is going to simplify we're going to be able to cancel and squared from both the numerator and denominator

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and this will simplify to 1 over n that means we're going to call our comparison series 1 over and so

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let's go ahead and say that this series from N equals 1 to infinity of 1 over n and we'll go ahead and

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call the original series a 7 and we'll call the comparison Series B 7.

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So now we want to do is show that both of these series are always going to be positive.

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So if we look at the original series we know we're starting at and equals 1 and we're counting up and

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equals 2 equals 3 and equals four.

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All the way up to infinity.

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Well if we plug in N equals 1 in the numerator here we can get 1 squared which is 1.

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Here we're going to get one plus two is three so we're going to end up with one third.

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If we plug in and equals 2 we're going to get 4 over 8 plus 2 is 10 for over 10.

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Or two over five.

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If we plug in N equals three we're going to get eight over twenty seven plus two is 29.

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So we're going to get 8 over 29 and we could keep going.

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Let's go ahead and look at though the comparison Series B 7.

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We're also starting at any one equals one we get 1 over 1 or 1 at N equals 2 we get 1 1/2 and equals

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3 we get one third and we could see that we could keep going.

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But in either case there's no value of.

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And then we could plug in if we're counting up from N equals 1 2 3 4 et cetera that we could plug into

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either series that would make any of these terms negative.

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And so we can say then that a 7 is always going to be greater than zero and we can say that B-7 is always

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going to be greater than zero.

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So those two conditions have been met and the next thing we want to show is that the limit is and goes

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to infinity of the ratio of these series is going to be greater than zero.

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So if we take the limit as and approaches infinity of a sob in our original series and squared over

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and cubed plus two and we divide it by our comparison series we're going to divide this whole thing

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by one over and.

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But keep in mind that that's exactly the same thing as multiplying by an over 1.

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The result then here is going to be the limit as and goes to infinity will multiply and squared by and

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we'll get and cubed over and cubed plus two.

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Now in order to take the limit what we want to do is we want to multiply both the numerator and denominator

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by the highest degree variables so we have one over and cubed and one over and cubed like this.

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What that's going to do when we say one over and cubed times and cubed we're going to get one in the

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numerator so we're going to have limit as and approaches infinity we're going to have one in the numerator

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and denominator one over and cubed times and cubed is going to give us 1.

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Then we're going to have here plus two over and cubed that way when we take the limit as and goes to

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infinity the denominator right here is going to get extremely large to divided by an infinitely large

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number is going to be zero.

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So this term is going to go to zero which means it's going to become zero there.

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And we're just going to end up with one over 1 or that this limit here if we call this limit L that

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L is going to be equal to 1 and one is greater than zero.

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So we've shown that this condition here has also been met because Al is greater than zero.

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So now that those conditions have been met we can go on to our conclusions.

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What we want to do is evaluate be stubborn and say whether B-7 converges or diverges.

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If we look here at B-7 we notice that it's the harmonic series one divided by and is a harmonic series

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and we know that the harmonic series diverges.

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So since B Subban diverges we're looking at this conclusion right here.

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If B-7 diverges then A7 also diverges.

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So since we've shown that B Subban diverges we can conclude that a Sabun also diverges and that's how

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you use the limit comparison test to say whether a series converges or diverges.