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₌₁∞ (xᵏ/kᵏ)

Accepted Answer

Hello. In this video, we are going to be finding the radius and the interval of convergence for the given power series. Now, the power series given to us is the infinite series starting at 1 of Y to the power of N divided by N to the power of 2N. Now, since both the numerator and denominator have some exponent of N, we can actually rewrite the current power series. We can represent the power series function. As the quantity y divided by n squared all raised to the power of n. Now, conveniently here, what we can go ahead and do is in order to find the radius of convergence, we can apply the root test. And the root test is going to ask us to take the limit as N approaches infinity of the nth root of the absolute value of our function. So what we're going to do is apply the root test in order to find the radius of convergence. So by using the root test, we will have the limit as N approaches infinity of the n root of the absolute value of Y divided by N2d raise to the power of N. Here, the nth root is going to cancel with the exponent of N, and we will be left with the limit as N approaches infinity of the absolute value of Y divided by n squared. Now, when we apply the limit, we are going to have Y divided by infinity, which is going to result in the limit of 0. Now, 0 tells us two things about the radius of convergence and the interval of convergence. When we apply the ratio or root test and we get a result of zero, we can immediately assume that the radius of convergence is going to be infinity. And because the radius of convergence is infinity, then the interval of convergence is going to be all real numbers from positive infinity to negative infinity, and this is going to be the solution to the given problem. With the overall solution being 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₌₁∞ (xᵏ/kᵏ)

Key Concepts

Power Series

Radius of Convergence

Interval of Convergence

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