Calculus 2: Direct Comparison Test

This article is a summary of the YouTube video ‘Direct Comparison Test – Calculus 2’ by The Organic Chemistry Tutor

Written by: Recapz Bot

Written by: Recapz Bot

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How does it work?
The video explains direct comparison test for series convergence/divergence with examples.

Key Insights

  • The video discusses the direct comparison test in sequences and series.
  • The direct comparison test states that if a big series (b sub n) converges, then the small series (a sub n) will also converge.
  • Conversely, if the small series diverges, then the big series will also diverge.
  • The reverse statements of the test are inconclusive.
  • Graphs are used to illustrate the concept of convergence or divergence based on the comparison of the big and small series.
  • Examples are given to apply the direct comparison test.
  • Example 1: Series of 1/(4+3^n) is compared to 1/3^n, showing convergence.
  • Example 2: Series of 1/(n^3+5) is compared to 1/n^3, again showing convergence.
  • Example 3: Series of 1/(4+n^(1/3)) is compared to 1/n^(1/3), proving divergence.
  • Example 4: Series of 2^n/(7^n+8) is compared to 2/7^n, demonstrating convergence.
  • Example 5: Series of ln(n)/n is compared to ln(n)/n^2, resulting in divergence.
  • The video emphasizes understanding the direct comparison test, and additional examples are encouraged to solidify the concept.

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Transcript

In this video, we’re going to talk about the direct comparison test. So let’s say if we have the sequence b sub n and it’s greater than or equal to a sub n, which is greater than 0. So b sub n, we can describe it as the big sequence and a sub n is the small sequence. And so the basic idea behind the direct comparison test, it says this, at least the first statement, if the series, if the big series converges, then the small series will converge as well. And then the second statement, if the small series diverges, then the big series diverges as well. And so that’s the basic idea behind the direct comparison test. Now the reverse statement is inconclusive. So I can’t say this, if the small series converges, then the big series converges. That’s not the case. The big series, it may converge or it may diverge. So it’s inconclusive. And the same is true for the reverse statement of statement number two. So I can’t say that if the big series diverges, then the small series will diverge as well. That’s not the case. So just keep that in mind.

Now let’s analyze this concept with a graph. So let’s say if the big series converges. So it becomes horizontal over time. What can you say about the small series? Well, a sub n is less than b sub n, so then it has to converge as well. And so that’s the basic idea behind the direct comparison test.

Now let’s say if the small series converges. The big series, does it have to converge or can it diverge? Well, it can do both. It may converge, we don’t know, or it may diverge. It’s inconclusive. And so this graph can help you to see the concept behind the direct comparison tests. So if the big series converges, the small series has to converge. And if the small series diverges, then the big series have to diverge. So make sure you understand that.

Now let’s work on some examples. So consider this problem. Let’s say if we have the series from 1 to infinity of 1 over 4 plus 3 to the n. Will the series converge or diverge?

So what can we compare this expression to? Which term is insignificant? When n becomes very large, the 4 is insignificant. So let’s drop the 4. If we do that, we’re going to get this. Now which term is larger? This one or this one? Notice that the denominator of this fraction is greater than this one. 4 plus 3 to the n is greater than 3 to the n. As you increase the denominator of a fraction, the value of the whole fraction goes down. So because the denominator has a higher value, the whole fraction has a lower value, which means that 1 over 3 to the n is greater than 1 over 4 plus 3 to the n. So this is the big function and this is the small function. So we can describe this as a sub n and b sub n. So this series is going to be smaller than this one. Now what can you tell me about this particular series? What type of series is it?

Now we can rewrite 1 over 3 to the n as 1 over 3 raised to the n. Notice that that is a geometric series with a common ratio of 1 over 3. And if the common ratio is less than 1, what do you know about the geometric series? The geometric series is convergent. Now if the larger series is convergent, what can we conclude about the smaller series? The smaller series must be convergent as well. So by the direct comparison test, we know that this series converges.

Now let’s try another example. Determine the convergence or the divergence of this series. Let’s say it’s 1 over n cubed plus 5. So when n becomes very large, the 5 is insignificant relative to n cubed. So we can compare this series to this one, 1 over n to the third power. Now which sequence is greater? 1 over n to the third plus 5 or 1 over n cubed? Now n to the third plus 5 is greater than n cubed. Because the denominator of this fraction is larger, the value of the whole fraction must be smaller. So 1 over n cubed plus 5 has to be less than 1 over n cubed. So we can put less than or equal to. So therefore, this is the small series and this one is the big series. So what do we know about this series? What type of series do we have? This is a p series. And so p is equal to 3. Now whenever p is greater than 1, the series converges. So in that case, if the big series converges, then the small series must converge based on a direct comparison test.

Let’s try this example. So we have the series from 1 to infinity, 1 divided by 4 plus the cube root of n. So use the direct comparison test to determine if it’s going to converge or diverge. So when n becomes large, the 4 is insignificant. So we can write this expression. And so this series is going to be bigger than the first one. So this is the b sub n and this is our a sub n. Now we can rewrite this series like this. The cube root of n is n to the 1 third. So basically we have another p series, but this time p is equal to 1 third. So therefore, because p is less than or equal to 1, we have a divergent series. So what can we say about this series? Is it convergent or divergent? So if the big series diverges, will the small series diverge as well? No, it’s inconclusive. So remember what the direct comparison test says. And that is, if the small series diverges, then the big series diverges well. But if the big series diverges, the small series, it doesn’t have to diverge. It can converge. So in this case, the direct comparison test is inconclusive. You may have to use the limit comparison test, which I’ll talk about that in another video.

Now let’s try another problem. Let’s say it’s 2 to the n over 7 to the n plus 8. So use the direct comparison test to determine if this series will converge or diverge. So what should we compare it to? All we need to do is get rid of the 8. And so we’re going to get this expression. Now relative to this fraction, we’ve increased the denominator by 8 on the left. So anytime you increase the denominator of a fraction, the value of the whole fraction goes down. So therefore, the series on the left is less than the series on the right. So this is our a sub n, and this is our b sub n. Now let’s rewrite this expression. So let’s change it to 2 over 7 raised to the n. So now it looks like a geometric series. And the common ratio is 2 over 7. So the absolute value of r is less than 1, which means that the geometric series converges. Now according to the direct comparison test, if the big series converges, then the small series converges as well. And so that’s the answer.

Here’s another problem that you could try. So it’s going to be natural log of n over n. Will that series converge or diverge? And what should we compare it to? So when n goes to infinity, n increases faster than the natural log of n. So the n on the bottom is significant. What can we compare ln n to? The natural log of n is greater than 1 whenever n is equal to or greater than 3. For instance, ln 2 is 0.693, so that’s not going to work. ln 3 is 1.0986. So when n is 3 or more, this statement becomes true. So this is going to be the big series, and this one is associated with the small series. So this is a harmonic series, or p-series, where p is 1. And if p is equal to 1, then the series is divergent. Now if the small series is divergent, what can we conclude about the big series? That means the big series has to be divergent. And so that’s it for this example.

Let’s try two more examples

This article is a summary of the YouTube video ‘Direct Comparison Test – Calculus 2’ by The Organic Chemistry Tutor