Практика Shadowing: Simple Guide to Series Convergence Tests - Изучайте разговорный английский с YouTube

So we are going to talk about what convergence tests to use

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So we are going to talk about what convergence tests to use

when we're trying to figure out whether an infinite series converges.

We'll go over when to use the test for divergence,

the geometric series, the p-series,

the comparison test, the limit comparison test,

the alternating series test, the integral test, and the ratio test.

All of that good stuff.

So I've broken down every infinite series we'll generally see on an exam into four different categories here,

and we're going to go over how to deal with each one.

Let's start out with the basic category.

This is talking about those series that we already know whether they converge or diverge right away.

And those two are going to be the geometric series,

the sum from n equals 1 to infinity of r to the n,

which we know converges when the absolute value of r is less than 1,

and the sum from n equals 1 to infinity of 1 over n to the power of p.

And that's going to converge when p is greater than 1.

So this is the geometric series and the p series formula.

When we see a series that we can write in terms of one of these two things,

we should always do that first because we can instantly get to an answer just like that.

But these aren't the most common types of series,

So let's take a look at some of the other categories.

The next one is almost basic series.

And these are series that are very,

very close to a geometric series or a p-series,

but are just not quite there.

There's a little something off.

For example, the sum from n equals 1 to infinity of 1 over the square root of n cubed plus 5.

So when n gets really,

really, really large, notice inside this square root,

n cubed is going to blow up.

It's going to be massive,

but 5 is just going to stay as 5.

So the bigger we get in this series,

the less this plus 5 is going to matter.

It's basically just going to be n cubed.

So we can write this as basically the sum from n

equals 1 to infinity of 1 over the square root of n cubed.

But notice that the square root of n cubed is something

that we can write as n to the power of 3 halves.

And if we have the sum from n equals 1 to infinity of 1 over n to the 3 halves,

that is a p series,

and we know that it converges.

So these kinds of series where it's basically a p series,

it's basically geometric, but there's a little thing that's throwing it off,

these are almost basic series.

series.

And the way that we evaluate this kind of series is always by using a comparison test.

So there are two comparison tests which are the standard comparison test,

or the direct comparison test,

and the limit comparison test.

And it doesn't generally matter which one we use,

the limit comparison test will usually work in more of the cases,

but sometimes we can get away with the comparison test as well.

So in the case of this series here,

notice on the bottom n cubed plus 5,

that's always going to be bigger than n cubed.

And if we have something bigger in the denominator,

that's going to make our results smaller.

So we know that 1 over the square root of n cubed plus 5,

that's always going to be less than 1 over the square root of n cubed.

Of course on an exam we'll have to prove this more rigorously,

but this is the general reasoning that we need.

Once we get to this,

we can use the direct comparison test to say,

well, the sum of 1 over the square root of n cubed,

that's a p series and that converges.

Because this is less, this must also converge.

So that's how we can use the comparison test or the limit comparison test to evaluate these almost basic series.

And we'll go through a few exercises at the end where we can practice identifying which type of series we're dealing with.

So the next category we're going to look at is alternating series.

And this is a very simple category.

It is exactly what it sounds like.

So if we have the sum from n equals 1 to infinity of negative 1 to the n times b sub n,

where b sub n is positive,

this is an alternating series.

And any time we have an alternating series,

it's always a good idea to try using the alternating series test to see whether it converges.

As an example,

if we had the sum from n equals 1 to infinity

of negative 1 to the n over the natural log of n plus 1,

this is an alternating series and we're going to try using the alternating series test on it.

In this case we can actually prove that this is an alternating series that converges by that test.

So there will be some cases where an alternating series,

we can't prove it converges by the alternating series test,

but it's always a good thing to try first if we can pull out a negative 1 to the n.

Now the last type of series we're going to talk about is the weird series.

And these are the series that don't really fall into any of the categories that we already talked about.

As an example,

we might have the sum from n equals 1 to infinity of n times e to the negative n squared.

So this clearly isn't one of our basic forms,

and it's not really almost basic either because not only do we have this n,

but it's not just e to the negative n,

it's e to the negative n squared and that's just strange,

it's not alternating either.

So this is a case where we're going to have to use a different method.

And there are two different methods that we use for weird series.

The first is the integral test.

And we use the integral test any time we can integrate the series.

So in this case, the sum from n equals one to infinity of n e to the negative n squared,

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well we are able to do the integral from 1 to infinity of x e to the negative x squared dx.

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And as long as the conditions for the integral test are satisfied,

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meaning it's continuous, positive, and decreasing over time,

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we are able to use the integral test for this kind of series.

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There's one other kind of weird series,

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which I'll give an example up here.

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For example,

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the sum from n equals 1 to infinity of n factorial

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times 3 to the power of 2n over 4 to the n times n plus 3 factorial.

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So in this case, the important thing about these kinds of weird series is we have powers and we have factorials.

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So I'll write that here.

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When we have powers and factorials,

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And when I say powers,

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I'm talking specifically about n in the power.

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In that case, we're going to use the ratio test to evaluate these.

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So in this case, if we use the ratio test for this series,

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we would be able to evaluate it to a particular number

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that we could use to prove that this either converges or diverges.

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So those are the two kinds of weird series we're talking about.

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If you can do the integral,

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do the integral test, as long as the conditions are satisfied.

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And if we have factorials and powers flying around everywhere,

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that's when we're going to try the ratio test.

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So now that we've identified all of these categories,

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let's go through a few exercises where we try to figure out

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which type of series we're looking at so we know which test to apply.

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Let's start out with a basic one,

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the sum from n equals 1 to infinity of 5 halves to the power of negative 2n.

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This already sort of looks like a geometric series, right?

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So we have something and it's sort of to the power of n.

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So before we look at anything else,

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let's see if we can interpret it as just this basic form of a geometric series.

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To do that, we want to take a look at the exponent first.

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because we know that if we have 5 halves to the negative 2n,

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we can actually split up the powers as 5 halves to the power of negative 2

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and then to the power of n.

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And once we do this,

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we can rewrite the series in a form that we like a lot more.

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We'll have the sum from n equals 1 to infinity.

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Well, what's 5 halves to the negative 2?

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That's going to be the same thing as 2 fifths squared, right?

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We have to flip it over for a negative power.

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that's going to be 4 divided by 25, right?

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2 squared over 5 squared,

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and then we raise that to the power of n.

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And now this is exactly in the form of a geometric series,

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r to the n, and our r is 4 over 25.

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So if we have our ratio being 4 25ths,

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that's less than 1, which means we have proved that this series converges by the geometric series.

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Next up,

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let's take a look at the sum from n equals 1

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to infinity of n squared times 3 to the n over n factorial.

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Notice in this case, this is already looking kind of weird.

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So we might be looking at this weird category to start out.

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And we also notice that we have factorials and we have numbers raised to the power of n.

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And those two things indicate to us that we should probably be using the ratio test.

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So in the case of this series,

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we won't go through all the work right now because we're just trying to identify which test to use,

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but if we use the ratio test on this,

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we would be able to prove that the common ratio as n approaches infinity is less than 1,

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and therefore that this series converges by the ratio test.

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The next one we're going to take a look at will be a little different.

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It'll be the sum from n equals 1 to infinity of

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n to the sixth plus n squared over n to the seventh plus 3.

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Now, notice in this case,

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as n gets really, really large,

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n to the sixth is going to be much,

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much, much bigger than n squared,

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and n to the seventh is going to be much,

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much bigger than 3,

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which means

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that really what we're looking at is very similar to the

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sum from n equals 1 to infinity of n to the sixth over n to the seventh.

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But we can write this,

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n to the sixth over n to the seventh,

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we put the powers together,

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as just 1 over n.

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So we can say that our series that we have here is very much like 1 over n,

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as n gets very large.

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And this is a type of p-series.

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So in reality, the series that we're looking at here is an instance of an almost basic series.

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So we're going to want to use either the comparison test

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or the limit comparison test to turn what we have here into the p-series.

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Now because we have this n squared on the top,

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using the comparison test is going to be a little strange,

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so I think it would be easier in this instance to use the limit comparison test.

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we would use the limit comparison test to compare this series to the sum of 1 over n.

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And then we would get the value of 1 as our answer

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and because 1 is the result that we get and the sum of 1 over n diverges,

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the harmonic series, p is less than or equal to 1,

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then we know that our original series diverges as well.

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Next up,

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let's take a look at the sum from n equals 1

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to infinity of negative 1 to the n over n factorial plus n squared plus 3.

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Now you might be tempted to look at this as a weird series because of the n factorial,

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and that might work, but also notice that we have a negative 1 to the n in the numerator here,

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which means this is an alternating series.

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the first thing we always want to try is the alternating series test.

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And in this case, we would actually be able to prove that this sum converges by the alternating series test.

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So we're going to look at one more series here,

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which is the sum from n equals 1 to infinity of 1 over the inverse tangent of 6n.

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Now this is not a basic series.

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It's not almost basic either because we have this inverse trig function kind of strange.

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It's not alternating and we might be tempted to look at this as a weird series,

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but we should hold on first and examine what this series actually looks like.

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If we think about the inverse tangent function,

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what would the graph of that function look like?

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Well, as n on this x-axis here increases,

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the inverse tangent is going to look like this.

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It's going to have a vertical asymptote at the value of pi over 2,

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which means if we divide by the inverse tangent,

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this is not going to go to 0.

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So if we take the limit as n approaches infinity of the inside of the sum,

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1 over the inverse tangent of 6n,

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that result is not going to be zero.

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And if this limit is not equal to zero,

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then by the test for divergence, this series must diverge.

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So this exercise is a quick reminder that any time we're doing series,

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no matter what it looks like,

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we always have to remember the test for divergence,

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because it might seem that it would be a good idea to try some other series test,

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but as long as we can prove that the inside of the sum never goes to zero.

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We can get off really easy just doing a limit and proving that it diverges.

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So, these are the four categories that it's always good to remember

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when you're trying to figure out which convergence test to use.

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If you see a sum that you can write in the form of sum number to the power of n,

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that's a geometric series, so that's a basic form that you can evaluate directly.

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The same goes for anything you can write as 1 over n to the power of p.

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The next category is things that are almost these basic forms.

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For example, the sum of 1 over the square root of n cubed plus 5.

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If we could take out that 5,

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then we would get the n to the p that we want.

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So this is an almost basic sum,

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and these are where we use the comparison

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and the limit comparison to write this sum as we say it's very similar to another form

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that we know how to evaluate.

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The next category is alternating series.

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Anytime a series is alternating,

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the first thing we want to try is the alternating series test

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to see if we can use that to prove that it converges.

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And the final category is the weird series.

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So they aren't basic, they aren't almost basic,

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and they're not alternating either.

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When we get these kinds of series,

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depending on the form, we're either going to use the integral test if we can do the integral,

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or we'll use the ratio test if it has things to the power of n and factorials.

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And the last thing to remember is the test for divergence.

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Anytime the series isn't going to go to zero,

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we can just take that limit,

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show it's not zero, and then we're done.

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