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∞ sinᵏ(1/k) xᵏ

Accepted Answer

Hello. In this video, we are going to be finding the radius of convergence and the interval of convergence for the given power series. The power series given to us is the following. We have the infinite series starting at 1, and we are given tangent to the power of N of 1 divided by N2d, multiplied by Y to the power of N. Now the first thing to notice in this power series is that both of the terms are raised to the power of N. So we can go ahead and rewrite the power series and represent this as one entire function raises to the power of N. So we are going to have tangent of 1 divided by N squared, multiplied by Y, and this will all be raised to the power of N. Now, because this entire function is raised to the power of N, we can reproach this problem by using the root test. The root test asks us to evaluate the series by taking the limit as n approaches infinity of the nth root of the absolute value of a sub n. So what we're going to do is we're going to take the absolute value of our current series and take the end root of the absolute value. So by the root test, we will have the limit as N approaches infinity of the nth root of our function. Tangent Of 1 divided by n squared. Multiplied by y, this is all absoluted and raised to the power of n. This is going to further simplify to leave us with the limit as N approaches infinity of the absolute value of tangent of 1 divided by n2d multiplied by the absolute value of Y. Now, because we are taking a very large value of N, For a large value of n. Tangent of 1 divided by N2 is going to approximate to be 1 divided by N2. And so what this means is that when we apply the limit, or we can replace this function of tangent with one divided by N2d. So we will have the limits as N approaches infinity of the absolute value of 1 divided by N2, multiplied by Y. But once we apply the limit, we're just going to get the output of 0. Now because we got the output of 0, there are two things that we can assume right away about our radius of convergence and our interval of convergence. If the output of the test is zero, then both the radius of convergence and the interval convergence are going to be infinity. So this is going to be the solution to this given problem. And the overall solution of the problem is going to be D. So I hope this video helps you in understanding on 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∞ sinᵏ(1/k) xᵏ

Key Concepts

Power Series and Convergence

Radius of Convergence

Interval of Convergence

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