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Today we're going to be talking about how to use the term test to try to show whether a series diverges.

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Now keep in mind that the term test is also called the divergence test.

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And you also hear it called the zero test.

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It's all the same test.

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It's just called different things so you'll see it by different names.

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Just know that it's the same test here.

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And in this particular video we're going to be doing three separate examples I've written three different

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infinite sums here and we're going to be using the divergence test to test each one of them for divergence.

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Now here's what the divergence test says and I like to call it the divergence test as opposed to the

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term test or the zero test because it helps remind me that this test is really only good for talking

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about divergence.

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It tells us nothing about whether or not a series converges.

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So here's what the test says.

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It says that if we take any series and we'll call it a 7.

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And any of these theories that we have here can be a sudden we can call this a seben here this natural

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log of this rational function here could be a 7.

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Or this quantity here this series could be a subset.

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So any of these series if we have a series a seven and we take its limit and goes to infinity if that

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infinite limit is not equal to zero if the result of that limit is not equal to zero then this test

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tells us that the series diverges.

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That doesn't mean that if this is equal to zero that the series converges.

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It just means that if the limit is equal to zero that the test is inconclusive so the test really only

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can show us whether or not a series diverges.

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If the limit is and goes to infinity of the series is not equal to zero.

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Then we've proven that the series diverges.

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But if it is equal to zero.

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We haven't shown that the series converges.

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This particular test is just inconclusive.

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We can't get any information from the test and we would have to use a different test to see whether

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or not the series converges or diverges so that's why I like to call it the divergence test because

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it helps remind me that we're only making conclusions about divergence that we can't say anything about

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whether or not a series is convergent.

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So let's go ahead and take a look at our first example the infinite sum from N equals 1 to infinity

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of and minus 1 over 3 and minus 1.

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So what we're going to do to use the divergence test to test this series divergence we're going to say

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the limit as an ant goes to infinity.

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Of a sudden which is.

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In this case and minus one over three and minus one.

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We're just going to evaluate this and see what the result is.

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Now in this case it's really easy because we just have a rational function and remember that when we

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have a rational function we can evaluate this infinite limit in one of two ways.

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We can either look at the coefficients on the largest degree n terms in both the numerator and denominator

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so the largest degree term in the numerator is this and variable right here this and term.

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The largest degree turn the denominator is 3 n.

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So you basically have 1 and over 3 n.

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And if we look at these coefficients here then we know that the limit is one third.

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The other way we can do it is by multiplying both numerator and denominator by 1 divided by the largest

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degree the highest degree and variable.

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If we had an n squared anywhere in this rational function we would have to multiply by 1 over and squared

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divided by 1 over and squared but N itself and to the first power is the largest degree a value of and

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that we have.

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So we multiply by 1 over and divided by 1 over 10.

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And what that gives us is an at times 1 over N is just one negative 1 times 1 over n is minus 1 over

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N divided by 3 End Times 1 over and the ends cancel.

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We're just left with three minus one over end because negative 1 times one over end is negative 1 over

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N..

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Now if we take the limit as and goes to infinity of this function here what we'll get is these little

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fractions one over and negative one over in these become zero.

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Because when we have a small constant like this or really any constant divided by an infinitely large

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value that value that whole fraction itself goes to zero and therefore as you can see what we're left

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with is just this one third value here.

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So we know that the limit as and goes to infinity of our series A seben is equal to one third because

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one third does not equal zero.

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The divergence test by the divergence test we can conclude that this series a sub in a sudden diverges

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series diverges.

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So the divergence test is conclusive because this ONE-THIRD value is not equal to zero.

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OK so that was a sub n Let's go ahead and call this series B sidebands so we can distinguish between

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them and we'll call this one see.

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So if we do another example if we look at B Subhan here we're going to take the limit as and goes to

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infinity of the natural log of end squared plus 1 divided by 2 and squared plus 1 and using limit laws

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we can say that this limit is actually the natural log of the limit as and goes to infinity and squared

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plus 1 divided by 2 and squared plus 1.

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And from here we have the same type of rational function that we had in our first example and what we're

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going to get is just the coefficients of the highest degree and values.

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Here we get one and squared over two n squared.

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Everything else will cancel and we're just left with this one half these two coefficients on our end

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squared terms.

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The limit as and goes to infinity of 1 a half is just one half.

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So we get the natural log of one half.

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This of course is not equal to zero and therefore by the divergence test our series B saw then diverges

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Now if we look at our third example here because we have the summer of two separate terms like this

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we can break apart each term put it in its own.

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Some you can always do that if you have two terms like this added together or subtracted from one another

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you can break them apart so we'll say this some from N equals one to infinity of one over to the N plus

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the sum from an equals 1 to infinity of 1 over n times and plus 1.

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Now we can test these separately for divergence.

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So we'll test this first term here one over either the N will stay the limit as and goes to infinity

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of one over E to the n.

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Well if we evaluate this at some very very large number for N we can get some extremely large number

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and an infinitely large number here in the denominator and when we just have a constant divided by an

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infinitely large number this is going to go to zero.

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This is going to be equal to zero or by the divergence test remember the divergence test is inconclusive.

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If the limit is and goes to infinity is equal to zero.

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So the divergence test for this particular term is inconclusive.

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We would actually have to test it a different way.

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And what we would find if we rolled out the first several terms of this series is actually that it's

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a geometric series and we could use the geometric series test to test this for convergence.

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Let's go ahead and look at the second series here.

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One over end times the quantity and plus one will say the limit as it goes to infinity of one over.

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And at times the quantity and plus one.

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Again if we just plug in a very very large value for n we'll get an infinitely large value in the denominator.

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We've got a small constant here in the numerator.

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This value is also going to tend toward zero.

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We're just going to get zero for that value and therefore the divergence test for this particular series

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is also inconclusive.

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The way that we would test it is we use partial fractions to break apart this series here.

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What we'd end up with is one over n minus one over and plus 1 which would replace the series we have

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here.

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And then if we write out the first several terms of this series we realize that it's a telescoping series

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and treating it as a telescoping series it becomes very easy to find the sum of this series and therefore

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to conclude that it converges.

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So that's a couple examples of how we use the divergence test to determine whether or not a series diverges

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or just to say that the divergence test is inconclusive and use a different test instead.