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.

−x²/1 + x⁴/2! −x⁶/3! + x⁸/4! − ⋯

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 we are given an expanded power series. It is X cubed minus X 6 divided by 2 factorial, plus x 9th divided by 3 factorial, minus X 12th divided by 4 factorial, plus an infinite amount of sums afterward. Now what we need to do is we need to go ahead and represent this in summation notation. Now, what we have here is an alternating series with factorials in the denominator, and a numerator that can be represented as X to the power of 3 N. And so what this means is that we can rewrite the expanded series to be the infinite series starting at 1 and going to infinity of -1 to the power of N + 1. Multiplied by X to the power of 3 N divided by N factorial. Now, just by taking a look at this given series, because we have an alternating term in the numerator, An exponent N in the numerator and an N factorial in the denominator, we can find the radius of convergence by using the ratio test. By using the ratio test, we need to take the limit as N approaches infinity of the absolute value of the quantity A N + 1 divided by AN. So we're going to take the absolute value of our function, plug in the value of M + 1, and divide by the original function. Now, here, by taking the absolute value, the absolute value is going to eliminate any alternating terms within our series. So we are going to have the limit, as N approaches infinity of the absolute value of X to the power of 3 multiplied by M + 1, divided by N + 1 factorial. Now, here, we will need to divide by the original function without the alternating term, but that's going to cause a complex fraction. So equivalently, what we can do is we can multiply by the reciprocal of our original function without the alternating term. So that is going to be in factorial, divided by X to the power of 3 N. Now, let's go ahead and simplify these terms. By taking a look at the X terms in the numerator, we have X to the power of 3, multiplied by M + 1, divided by X to the power of 3N. By distributing that 3 in the exponent, we get X to the power of 3N + 3 divided by X to the power of 3N. Then what we can do is we can express this numerator as a product of exponents, and rewrite the numerator as X to the power of 3N multiplied by X cubed, divided by X to the power of 3N. And this is going to further simplify, to leave us with X cubed. So that's going to be the X terms simplified. But what about the N terms? In the numerator, we have N factorial, and in the denominator, we have M + 1 factorial. Now, if we were to expand the factorial in the denominator, we can rewrite the denominator to be N +1, multiplied by N factorial, which is going to simplify to be 1 divided by N + 1. So by applying these simplifications, we can further rewrite the limit to be the limit as N approaches infinity of the absolute value of X cubed divided by N + 1. Now, here, if we were to apply the limits, we would end up with X cubed divided by infinity, and anything divided by infinity is going to result in 0. Now because we have got a value of 0. In the ratio test, this tells us two things about the radius of convergence and the interval of convergence. A result of zero tells us that we have an infinite radius of convergence, and because we have an infinite radius of convergence, the interval convergence is going to be all real numbers from negative infinity to positive infinity. And this is going to be the solution to the problem, with the overall solution being a. 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.

−x²/1 + x⁴/2! −x⁶/3! + x⁸/4! − ⋯

Key Concepts

Power Series

Radius of Convergence

Interval of Convergence

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