Worked example: interval of convergence

- [Instructor] So we have an infinite series here, and the goal of this video is to try to figure out the interval of convergence for this series. And that's another way of saying for what x values, what range of x values, is this series going to converge? And like always pause this video and see if you can figure it out. When you look at this series, it doesn't fit cleanly into something like a geometric series or an alternating series. When I see something like this, I think about the ratio test, because it tends to be pretty general. To apply the ratio test, we want to think about the limit, the limit as n approaches infinity of the n plus oneth term, divided by the nth term and the absolute value of that. If this thing is less than one, so when this thing is less than one, then we are going to converge. And the x values that make this thing greater than one, we are going to diverge. And the x values that make this equal to one, well then we're going to be inconclusive and so we're going to have to use other techniques to think about whether we're going to converge or diverge. So let's just think about this, let's just evaluate this, so let's do that. Limit as n approaches infinity of the absolute value of a sub-n plus one well that's going to be x to the n plus one. Let me color code this, just so we know what we're doing. So this thing right over here is going to be x to the n plus one over n plus one times five to the n plus one. We're going to divide that by the nth term. We're going to divide that by the nth term. And that's just going to be x to the n over n times five to the n. We're going to take the absolute value of this whole thing. Now let me simplify this. I'll simplify it down here. So this is the same thing as x to the n plus one over n plus one times five to the n plus one, times the reciprocal of this. So it's going to be n times five to the n, over x to the n. And we could simplify this. This is going to be equal to, let's see, you divide numerator and denominator by x to the n, you're left with just an x. Then divide numerator and denominator by five to the n. That is going to be, that's a one, this is a one and then this is just going to be, five to the n plus one divided by five to the n, that's just going to be a five. And so what do we have? We have x times n. x times n, over, distribute the five, five n, five n plus one. Oh, let me be careful there. Let me distribute the five. Five n plus five, five times n, five times one. Five n plus one. Five n plus five (chuckles). All right, so let me just rewrite that. This is going to be equal to the limit as n approaches infinity of the absolute value of this thing. And actually to help us with this limit, let me rewrite it a little bit. Let me divide the numerator and the denominator both by n. I'm not changing the value, I'm doing the same thing to the numerator and the denominator. I'm dividing it by the same value. So if I divide the numerator and the denominator by n, this is going to be the same thing as x over five plus five to the n. So when you divided the numerator and the denominator by n, it becomes very clear what happens as n approaches infinity, As n approaches infinity, x doesn't change, five doesn't change, but five over n goes to zero. And so this limit is going to be equal to x over five. So that's a pretty neat, clean thing. Now we can use this to think about, and actually let me write this. This is going to be the absolute value of x over five. Now we can think about under which conditions is the absolute value of x over five going to be less than one and we're definitely gonna converge? Under what conditions are we going to be greater than one and definitely diverge? And then under what conditions is it inconclusive? So let's just see when we know we can converge. So the absolute value of x over five is less than one. This is our convergence situation. Well, that's the same thing as saying that x negative one is less than x over five, which is less than one. And you multiply all the sides by five. This is the same thing as negative five is less than x, which is less than five. So if we know that this is true, this is definitely going to be part of the interval of convergence. We know that if x meets these constraints, then our series is going to converge. But we're not done yet. We have to think about the inconclusive case. So let's think about the scenario where the absolute value of x over five, absolute value of x over five is equal to one. Or another way of thinking about this, that means x over five is equal to one or x over five is equal to negative one. And this means that x is equal to five or x is equal to negative five. So these are the two inconclusive cases using the ratio test. So let's test them out individually by looking back at the series and just substituting x equals five or x equals negative five. So in the first scenario, let me find a new color here, let me use red. So the first scenario of x equals five, let's go to our series. Then the series is going to be the sum from n equals one to infinity of five to the n over n times five to the n. Well, this is just going to be equal to the sum n equals one to infinity of one over n. This is a harmonic series. This is the p-series where p is equal to one. And we know our p-series of p is equal to one. That's going to diverge. And we know the harmonic series we've done in other videos, this definitely diverges. So this diverges. You could do that by p-series convergence test. If the p for a p-series is one, well you're gonna diverge. So now let's think about, so five is definitely not part of our interval of convergence. Now let's think about x equals negative five. When x equals negative five, let me get a another color going here. When x is equal to negative five, then this thing is going to be equal to the sum from n equals one to infinity of negative five to the n. Actually let me just write that as, I'll write it out negative five to the n over n times five to the n. This is the same thing as the sum from n equals one to infinity. We could write this as negative one to the n times five to the, times five to the n, over n times five to the n. And now this thing, this is an alternating harmonic series. And so you could actually use the, you might already know that that converges, or you could use the alternating series test. The alternating series test, it might be a little bit clearer if I write it like this. That this is an alternating series. So in an alternating series test, if we see that this thing is monotonically decreasing and the limit as n approaches infinity is zero, this thing converges. The alternating harmonic series actually converges. So this converges. So given that this converges, you could view this as this boundary here. We would include that in our interval of convergence. So x doesn't just have to be strictly greater than negative five, it could be greater than or equal to negative five, but it has to be less than five. This is our true interval of convergence.