Radius and interval of convergence Use the Ratio Test or the R...

Question

Radius and interval of convergence Use the Ratio Test or the Root Test to determine the radius of convergence of the following power series. Test the endpoints to determine the interval of convergence, when appropriate.

∞
Σ (x/9)³ᵏ
k = 0

∞

Σ (x/9)³ᵏ

k = 0

Accepted Answer

Hi, everyone. Let's take a look at this practice problem. This problem says to determine the radius R and interval of convergence for the series, the sum of K equal to 0 to infinity of the quantity of Z divided by 4 in quantity rates to the quantity of 4K in quantity, using the root test. So, we're asked to find the radius of convergence, as well as the interval of convergence for our series using the root test. So we call for the root test that we need to calculate the limit L and L is just equal to the limit, as K approaches infinity. Of the Caith root. Of the absolute value of a sub K. Where a sub k is just going to be equal to the quantity of Z divided by 4 in quantity rates to the quantity of 4 K in quantity. So, substituting in this expression for a K into our expression for L, we see that L is going to be equal to the limit, as K approaches infinity of the quantity of the absolute value of the quantity of Z divided by 4 in quantity, raised to the quantity of 4 K. In quantity, and that is raised to the quantity of 1 divided by k in quantity. So here we've just rewritten the k through as raising that quantity to the power of 1 divided by k. So simplifying this expression, we see that this is going to be equal to the limit as k approaches infinity of the absolute value. Of the quantity of Z divided by 4 in quantity raised to the 4th power in quantity. So, if we look at our expression here, we see that the dependence on K has gone away. So, when we take the limit as K approaches infinity, we just get that this is going to be equal to the absolute value of the quantity of Z divided by 4 in quantity raised to the 4th power. Now recall that for this series to converge that L has to be less than 1. So that means that the absolute value of the quantity of Z divided by 4 in quantity raised to the 4th power has to be less than 1. Now, if we take the 4th root from both sides of this expression, we see that the absolute value of the quantity of Z divided by 4 in quantity has to be less than 1, and multiplying both sides by 4, we see that the absolute value of Z has to be less than 4. So that means that our radius of convergence R is going to be equal to 4. So that is the answer to the first part of this problem. Now, for the second part, we need to find the interval of convergence. So, based on the fact that the absolute value of Z has to be less than 4, that means that minus 4 has to be less than Z, which has to be less than 4. So what we need to do is to test the end points of this info to see if our series converges at those end points. So, we're gonna look at the case when Z is equal to -4. And for a series we'll have the sum of K equal to 0 to infinity. Of the quantity of -4 divided by 4. In quantity, raised to the quantity of 4k. And this is just equal to, if we subplify this expression, the sum of K equal to 0 to infinity of the quantity of minus 1 in quantity rates to the quantity of 4k. Now 4K is an even number. So here we have -1 being raised to an even power, which is just going to be equal to 1. So that means that this is going to be equal to the sum of K equal to 0, to infinity of 1, and we know that this series diverges. So if we just add up 1 a bunch of times as K approaches infinity, we're going to get that that series approaches infinity, so it diverges. Next, when you look at the case when Z is equal to 4, and we'll see something similar happening. So, we'll have the sum. Of K equal to 0 to infinity of the quantity of 4 divided by 4 in quantity raised to the quantity of 4k, and this is equal to the sum of K equal to 0 to infinity of 1 raised to. The quantity of 4 k in quantity, which is just equal to the sum of k equal to 0 to infinity of 1, which again this series diverges. So that means that neither of our end points belong to our interval of convergence. So that means that our interval of convergence is the open interval from -4 to 4. So that is going to be the answer to the second part of this question. So just to recap, we found that the radius of convergence R is equal to 4, and that our interval of convergence is the open interval from -4 to 4. So those are the answers that we're looking for for this problem. So I hope that this explanation has been useful, and I'll see you in the next video.

Radius and interval of convergence Use the Ratio Test or the Root Test to determine the radius of convergence of the following power series. Test the endpoints to determine the interval of convergence, when appropriate.

∞
Σ (x/9)³ᵏ
k = 0

Key Concepts

Power Series and Convergence

Ratio Test and Root Test

Interval of Convergence and Endpoint Testing

Watch next

Radius and interval of convergence Use the Ratio Test or the Root Test to determine the radius of convergence of the following power series. Test the endpoints to determine the interval of convergence, when appropriate.∞Σ (x/9)³ᵏk = 0

Key Concepts

Power Series and Convergence

Ratio Test and Root Test

Interval of Convergence and Endpoint Testing

Watch next

Radius and interval of convergence Use the Ratio Test or the Root Test to determine the radius of convergence of the following power series. Test the endpoints to determine the interval of convergence, when appropriate.

∞
Σ (x/9)³ᵏ
k = 0