Radius and interval of convergence of power series
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Power series intro
We've already seen many examples of infinite series. But what's exciting about what we're about to do in this video is we're going to use infinite series to define a function. And the most common one that you will see in your mathematical careers is the power series. And I'm about to write a general case of the power series. So I could imagine a function, f of x, being defined as the infinite sum. So going from n equals 0 to infinity of a sub n-- so a sub n is just going to be the coefficient on each term-- times our variable x minus some constant c. You could almost imagine this is shifting our function to the n-th power. So if I were to expand this out, I have my first term's coefficient, a sub 0, times x minus c to the 0-th power, plus a sub 1 times x minus c to the first power. This one, of course, will simplify just a sub 0. This would simplify to a sub 1 times x minus c plus a sub 2 times x minus c squared. And I could just keep going on and on and on. Now, when you see this, you might say, aren't our geometric series, don't those look like a special case of a power series if our common ratio was an x instead of an r in that case, or if our common ratio was a variable, I guess I could say? And you would be right. That absolutely would be the case. So a geometric series. So let's just think about defining a function in terms of a geometric series. And of course, we don't have to use x all the time as the independent variable, but this is kind of the most typical convention. I guess we could also use r as an independent variable if we wanted as well. But let's imagine a function g of x. We could have g of r if we wanted, but let's do g of x is equal to the sum from n equals 0 to infinity of a times x to the n. So this is kind of a typical geometric series here. And what's the difference between this and this? Well, the difference is is here, for every term we're going to have the same coefficient a, while over here we have a sub n. We're multiplying by a different thing every time up here. We're multiplying by the same thing over here. And in this case, this particular geometric series I just made, instead of having x minus c to the n, we have just x to the n. So you could say, well, this is a special case when c is equal to 0. And we can expand it out. This is a times x to the 0, which is just going to be a, plus a times x to the first, plus a times x squared. And we just go on and on and on forever. Now, what's exciting about this is we know that this, under certain conditions, will actually give us a finite value. This will actually converge. This will actually, I guess, give us a sensical answer. So under what conditions does that happen? Well, this converges if each of these terms gets smaller and smaller and smaller. And each of these terms gets smaller and smaller and smaller if the absolute value of our common ratio is less than 1. So let me write that down. So this converges if the absolute value of our common ratio is less than 1. Or another way of thinking about it, this is another way of saying that x is in the interval between-- it's less than 1 and it is greater than negative 1. And this term right over here, now x is a variable. x can vary between those values. We're defining a function in terms of x. We call this the interval of convergence. And so we know that if x is in this interval, this is going to give us a finite sum. And we know what that finite sum is. It's going to be equal to-- if it converges. So if it converges, this is going to be equal to our first term, which is just a-- this simplifies to a right over here-- over 1 minus our common ratio. What's our common ratio? Our common ratio in this example is x. Going from one term to the next, we're just multiplying by x. We're just multiplying by x right over there. Now, this is pretty neat, because we're going to be able to use this fact to put more traditionally-defined functions into this form, and then try to expand them out using a geometric series. And this whole idea of using power series, or in this special case, geometric series to represent functions, has all sorts of applications in engineering and finance. Using a finite number of terms of these series, you could kind of approximate the functions in a way that's simpler for the human brain to understand, or maybe a simpler way to manipulate in some way. But what's interesting here is instead of just going from the sum to-- instead of going from this expanded-out version to this kind of finite value, we're now going to start being able to take something in this form and expand it out into a geometric series. But we have to be careful to make sure that we're only doing it over the interval of convergence. This is only going to be true over the interval of convergence. Now, one other term you might see in your mathematical career is a radius. Radius of convergence. And this is how far-- up to what value, but not including this value. So as long as our x value stays less than a certain amount from our c value, then this thing will converge. Now in this case, our c value is 0. So we could ask ourselves a question. As long as x stays within some value of 0, this thing is going to converge. Well, you see it right over here. As long as x stays within one of 0. It can't go all the way to 1, but as long as it stays less than 1, or as long as it stays greater than negative 1. It can stray anything less than one away from 0, either in the positive direction or the negative direction. Then this thing will still converge. So we could say that our radius of convergence is equal to 1. Another way to think about it, our interval of convergence-- we're going from negative 1 to 1, not including those two boundaries, so our interval is 2. So our radius of convergence is half of that. As long as x stays within one of 0, and that's the same thing as saying this right over here, this series is going to converge.
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