Radius and interval of convergence Determine the radius and in...

Question

Radius and interval of convergence Determine the radius and interval of convergence of the following power series.

∑ₖ₌₀∞ (x/3)ᵏ

Accepted Answer

Hello. In this video, we are going to be finding the interval and radius of convergence for the given power series. Now the power series given to us is the infinite series starting at 0 of the function Y divided by 5, raise the power of N. Now, here what we have is a geometric series for the power series. This is a geometric series where the ratio is equal to Y divided by 5. Now for the convergence of the of the ratio of the geometric series. In order for a geometric series to be convergent, the absolute value of R must be less than 1. And so what this means is that if we want this ratio or this geometric series to be convergent, then the absolute value of Y divided by 5 must be less than 1. Now, solving for Y in this case is going to give us the absolute value of Y is less than 5. And what this tells us is that the radius of convergence is going to equal to 5. So this is our first solution to the problem. Now, by expanding this inequality, we are going to be told that Y is between -5 and 5. This is going to be our interval for the convergence, but we have to check the endpoints of the interval. In order to check the endpoints, we are going to take both of the endpoints of the interval and plug them back into the original series for the value of Y. Now, when Y is equal to -5, plugging this back into the original power series is going to give us -5 divided by 5, raise the power of N. And this is going to simplify to leave us with -1 to the power of n. Now, the problem with this is that this is an oscillating term, and because this term is oscillating, it will never converge to a finite value. Therefore, this endpoint is going to be divergent, so we will not include it in our interval of convergence. And now we need to check when Y is equal to 5. When Y is equal to 5, our series is going to simplify to be 5 divided by 5 rates to the power of N. This will further simplify to be 1 to the power of n. Which also further simplifies to just be the value of one. But this itself is a divergent series. And so what this means is that for the interval of convergence, the interval of convergence is going to be from -5 to 5, not including the endpoints, and this is going to be the second solution to our problem. So just to recap, our radius of convergence is 5, with our interval of convergence being from -5 to 5, not including the endpoints. And with that being said, the overall solution to this problem is going to be B. So I hope this video helps you in understanding how to approach this problem, and we will go ahead and see you all in the next video.

Radius and interval of convergence Determine the radius and interval of convergence of the following power series.

∑ₖ₌₀∞ (x/3)ᵏ

Key Concepts

Power Series

Radius of Convergence

Interval of Convergence

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