In this video, we're going to talk about how to determine if a series will converge or diverge. So, here's an example. Let's say if we have the infinite series of 2n. Will this series converge or diverge?
What would you say? Now, before we get into it, you need to know the difference between a sequence and a series. So in this example, the sequence a sub n is 2n.
So if you list out the terms of the sequence the first term a sub 1 is going to be 2 the second term a sub 2 is going to be 2 times 2 which is 4 the third is going to be 2 times 3 which is 6 and then 8 and so forth and so here we have a sequence of terms Now, S sub n relates to the partial sum. So, for example, S sub 1 is just a1, which is 2. S sub 2 is the partial sum of the first two terms. So, that's a sub 1. 1 plus a sub 2 which is equal to 6. s sub 3 is the sum of the first three terms which is 12 and s sub 4 is the sum of the first four terms which is going to be 12 plus 8 or 20. So if we want to we can write out a sequence of partial sums. So, S sub n, we can say S sub 1 is 2, S sub 2 is 6, S sub 3 is 12, S sub 4 is 20. And, if this sequence goes into infinity, this will keep on going forever. And so, this is termed an infinite series.
So this is equal to s sub infinity. It's the sum of the first infinite terms. Now how can we tell if it's going to converge or if it's going to diverge? How can we find the answer?
A more accurate way to describe this... is to say that this is equal to the limit as n approaches infinity of s sub n, which you could think of it as s sub infinity, but that might not be the most appropriate way to write it. So this means we're looking for the sum of an infinite number of terms. Now in order to determine if the series converge, you need to know if the sum exists.
Now let's go back to sequences. If you want to determine if the sequence is convergent or not, what you need to do is take the limit as n approaches infinity of the sequence a sub n, and if it's equal to a constant, then we say that the sequence a sub n converges. Now, if the limit increases or decreases without bound, or if it doesn't exist, then we say that the sequence a sub n, it diverges. Now, the same is true for a series, meaning the same concept is true. Now let's say if the infinite series or simply series, let's use a general sequence, a sub n, is equal to an actual number, which we'll call s.
So that's the sum of the infinite series. series is convergent it converges but now let's say if you can't determine the sum let's say if the sum goes to infinity or something positive or negative infinity then you can't really get a specific number there so therefore the series diverges So how can we determine the sum S? How can we know if it exists or not? Because if we can determine it, then we can determine if the series is going to converge or diverge. So what you need to do is find a general equation for the partial sums s sub n.
And it can vary depending on what situation you have. Once you have that, then you need to take the limit as n approaches infinity for s sub n. This is equal to the infinite series, the sum of the infinite series. And so if you can actually get a number for this value and prove that it doesn't equal infinity or it doesn't exist, if you can get a finite value as n approaches infinity for some general formula for S of n, then you can show that the series converges.
So let's start with this example. Determine if the infinite series converges or diverges. So, this particular series, does it equal infinity, or does it equal a number like 8?
If it equals a specific number, it's going to converge. If it equals infinity, it's going to diverge. Now, let's think about what this means. So, we know that a sub n is 2n.
And so if we list out the terms, the first term is going to be 2, the second term is 4, the third term is 6, and then 8, and then 10. Now, we have a summation notation, so we're adding these terms. And this will keep on increasing all the way to infinity. So just by looking at it, we know that the sum, the infinite sum, is infinity.
Just by looking at it intuitively. So that tells us that this series diverges. But let's see if we can confirm that answer another way.
Now, we need to determine a formula, S sub n, for this particular... series. Let's say if we wish to determine the partial sum of the series up to some value n.
How can we write a general formula for that? Well, you need to know what type of sequence you're dealing with. So here we're dealing with an arithmetic sequence.
And to find the partial sums of an arithmetic sequence, you could use this formula. It's a sub 1 plus a sub n divided by 2 times the number of terms. Now a sub 1 is the first term, which is 2. a sub n, well we have it here, it depends on what n is, so we're just going to write 2n.
and then there's an n out front. So 2 plus 2n divided by 2, that simply becomes 1 plus n, or n plus 1. So our general formula for the partial sums is n times n plus 1. Now, let's take the limit as n approaches infinity for the partial sums s sub n. So, that's the limit as n approaches infinity of n, n plus 1. So, what does that equal?
So, if you have infinity times infinity plus 1, what will you get? A large number times a large number will give you an even larger number. So this is going to equal infinity.
And so we could say that. The infinite series that we have here, let's use the general formula a sub n, that's equal to the limit as n approaches infinity for the partial sums s sub n. And we found that it's equal to infinity. So it doesn't equal a finite number. So because it equals infinity, that tells us that the series, original series, diverges.
And so that's a more systematic way of determining if the series is going to converge or diverge. Find the sum. If the sum is infinity, it diverges.
If it's equal to a finite number, it converges. Another test that is very useful that can help you to quickly tell if a series diverges is called the divergence test. And here's what it states.
So let's say if we have some sequence a sub n. If the limit as n approaches infinity of the sequence a sub n, if it does not equal 0, then the series, the infinite series, diverges. So what about if it equals 0? So if the limit as n approaches infinity for the sequence a sub n, let's say if it's equal to 0, then the series, it may diverge or it may converge.
We don't know. So we have to use other tests to figure it out. But the divergence test is a quick way to tell if it's going to diverge for sure or not.
If it does equal 0, then you need to use another test to see if it's going to converge or diverge. But if it doesn't equal 0, then automatically you know that it diverges. So let's go back to the first problem that we started with. Go ahead and use the divergence test to show that this series is divergent. Now the first thing you need to do is determine a sub n.
And so as you know, a sub n in this problem is simply 2n. Next, take the limit as n approaches infinity of a sub n. So the limit as n approaches infinity for 2n, does that equal 0?
It does not. 2 times infinity is equal to infinity. So, since the limit as n approaches infinity for a sub n, because it does not equal 0, then we can say that the series diverges.
And so, as you can see, it's a quick way to tell if it's going to diverge. If it did equal 0, then it may converge or diverge. Here's an example that you could try. Determine if the series converges or diverges.
So the first thing I'm going to do is start with the divergence test. So we could see that a sub n is equal to 5n plus 3 over 7n minus 4. So what is the limit as n approaches infinity for a sub n? What answer will you get? So let's begin by multiplying the top and bottom by 1 over n.
So we're going to have the limit as n approaches infinity, and this is going to be 5 plus 3 times 1 over n, divided by 7 minus 4 times 1 over n. Now keep in mind the limit... as n approaches infinity, for 1 over n, will always be 0. 1 divided by a large number will equal a small number. So this becomes 5 plus 3 times 0 divided by 7. minus 4 times 0. So this is equal to 5 over 7, which does not equal 0. So therefore, because the limit as n approaches infinity for a sub n, it does not equal 0, then we could say that the series, it diverges according to the divergence test.
Now let's think about what this means. As n approaches infinity, the sequence a sub n converges to 5 over 7. The sequence converges because the limit exists. But the series diverges.
because the sum at infinity doesn't equal a specific number. So in this problem, just keep in mind the sequence is convergent, but the series is divergent. So if we think about what this really means, as we approach...
an infinite number of terms, we're going to keep adding 5 over 7 as n increases. And so if you keep adding 5 over 7 when n is very large, the sum will continue to change. It's going to get higher and higher and higher.
So it doesn't stay at a specific value. So anytime the limit as n approaches infinity of a of n, if it doesn't equal 0, the sum at infinity will keep going higher and higher and higher. The only way for the sum to stay at a finite value is to keep adding 5 over 7. value is to add 0. Let's say the finite value is 100. Let me use an example. So let's say S of 1000 is 200. And let's say the limit as n approaches infinity for this sequence is 5. So let's say at a of 1000 it's also approximately 5. So s of 1001 will be 205 approximately. s 1002 would be 210. And so the sum will continue to go up as n increases.
So therefore this will never stop at a specific value. Now. Let's say, let me get rid of this stuff. Let's say if the limit as n approaches infinity for a sub n did equal to 0. Then, let's say a of 1000 is approximately 0. It might be like 0.001, but let's say it's close to 0. Then S of 1,001 is going to approximately be 200. S of 1,002 will still be 200. Because if you keep adding a small number to this, that keeps getting closer and closer to 0, it's going to stay at approximately 200. In that case, the sum converges.
And so that's why, in order for the series to converge, the limit as n approaches infinity for a series is 0. sub n so when you have a very high term it has to be close to 0 otherwise the sum will continue to increase so if you want to the Sun to stay at a finite value you have to keep adding 0 to it So in order for the series to converge, this must be equal to 0. But if it doesn't equal 0, then it has to diverge. If it does equal 0, keep in mind it may converge or it may diverge. The series doesn't have to converge, but it may converge, or the series may diverge.