In this video we will be investigating power series and finding the values of x for which this power series converges. So let's define what exactly is a power series. is a variable, then an infinite series in this form is called the power series. More generally, an infinite series in this form is called a power series centered at C, where C is a constant.
So just to tell you, in this form, it would say that the power series is centered at zero. You can tell if c is equal to 0, you would just get the x values. Because we are using x as our variable, we can write this power series as a function of x. The domain of f is the set of all x for which the power series converges, and that is our primary concern of this video.
Of course, every power series converges at its center because if you come back here and substitute in x is equal to c, you would, of course, get that series where all of those terms are 0, and our function would converge at that value. So C, whether it's 0 or some other constant, is always part of our domain, and so at the very least, our domain is a single point. But what else can happen? Our domain can be, of course, our single point of C.
It can also be an... an interval that is centered at C, it can also possibly be all real numbers. Our job today is to find out exactly what case we have.
And so from there, we have this theorem that says first of all, we're going to call R the radius of convergence. And so if we have a power series centered at C and R is the radius of convergence. Then the series converges at x is equal to c and our domain is only that. The series converges absolutely for when this is less than the radius of convergence and diverges everywhere else.
In that particular case, it is sometimes less than or equal to, sometimes not. You will always have to check the endpoints to know whether or not you include those endpoints or not. And then of course the third possibility is that the series always converges and our domain is all real numbers. So you're going to be faced with finding the interval of convergence for each series. In other words, when will this power series converge?
So you notice I've taken an Try to write out the first few terms so you can kind of get the idea of what you're looking at. So x, again, is a variable. Our job is to find out what values of x will make this converge. Let's investigate the graph of this infinite series.
When will this power series converge? The following animation will display these graphs. And as you watch it, see how these graphs converge between the values of negative 1 and positive 1. Here I am replaying the animation and letting n go from 1 to 100. The blue box in the middle is trying to emphasize how much those graphs are staying pretty much the same between the values of negative 1 and positive 1. But now let's look at the values when x equals to 1. Are those values converging?
No. In fact, as n is getting bigger, the value of x equals to 1 is unbounded. It just increases and increases. But what about when x is negative 1? Is it converging to a value?
As n gets larger and larger and larger, that value is getting more and more getting pretty much almost the same. In fact, right there it seems to stop, but N is still increasing. So you can see that these graphs converge somewhere between negative 1 and positive 1, and it looks like that it would include negative 1 but does not include positive 1. Now how are we going to show that with calculus? So how do we go about finding the interval of convergence?
Well, let's use what's called the ratio test. Recall the ratio test will converge if the absolute value of this ratio is less than 1. So that's what we're going to do. We're going to take the limit as n approaches infinity of the ratio of x to the n plus 1 divided by the square root of n plus 1 divided by this.
And remember when we divided we just multiplied by that reciprocal. What happens? Well, I can reduce this and this would be left with x. And so then I could say the limit as n approaches infinity of x times the square root of n. over the square root of n plus 1. What happens here as n approaches infinity?
Well again we have the same degree over the same degree, leading coefficients, so this This is going to reach a limit of 1. So this is going to be the absolute value of x. And you're going, well that didn't tell me anything. Yes but this did. Remember this will converge if that ratio is less than 1. So right there our radius of convergence is 1. In other words, this absolute value, remember when we remove the absolute value, we're going value, it means it's between the positive and negative value of that given number. Now the question is, is how do we arrive at knowing that we would include negative 1, but we don't include positive 1?
That's the tricky part. So we have to check the endpoints. So we're going to let x equal to negative 1. So we're going to go back to our series and let x equal to negative 1. So we're going to go back to our series and let x equal to Let X equal to negative 1. So we're going to have this series. 1, oh actually negative 1 to the nth power over the square root of n.
And the question is, does that converge or diverge? What test could I use? Well, because this is negative, being raised to a power, I know that's an alternating series, so I think I'll use the alternating series test.
So the first thing of that is to see if the limit of the nth term, does that equal to 0? Yes, it does. And the second part of the alternating series is to show that it decreases.
So is the n plus 1 term less than or equal to the nth term? And yes, that is true as well because we have a bigger denominator, so we have a smaller term. So we would say, yes, this converges. by the alternating series test.
So when x is equal to negative 1, this series converges, and so that's why I need to include negative 1. Now we have to let x equal to positive 1. So where there was an x, we put in a 1. Well, 1 to any power is just 1. and that's the same as 1 over n to the half power, you should recognize this as a p-series, where p equals 1 half. Since that's less than 1, that diverges. And so that's why we are not going to include positive 1. So our domain at which this series converges is negative 1 to positive 1, and it includes... negative 1, which we clearly saw in the graphical animation.
So you can write your answer like this or you can write it in interval notation with the brackets and the parentheses. Either way is okay.