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∞ (2x)ᵏ/k!

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. Now, the power series given to us is an infinite series starting at 0 of the function 3 Y raise the power of N divided by N factorial. Now, in order to find the radius of convergence, we can go ahead and apply the ratio test. The ratio test asks us to take the limit, as N approaches infinity of the absolute value of A N + 1 divided by AN. So in order to use the ratio test, we are going to plug in the quantity of M +1 into our original argument of the series, and then we will divide that by the original argument. So by applying the ratio test, we will have the limit as N approaches infinity of the absolute value of 3 Y, raise the power of N + 1 divided by the quantity N + 1 factorial. Now, here, we are going to have to divide by the original argument that was given to us, but that's going to create a complex fraction, so equivalently, what we can do is we can multiply by the reciprocal of the function that is given to us in the series. That reciprocal is going to be in factorial, divided by the quantity 3y raised to the power of N. And now what we're going to do is we're going to simplify. Let's go ahead and take a look at the three Y terms first. Now, if we take a look at the 3 Y terms in the numerator and denominator, we have 3 Y, raise the power of N + 1, divided by 3Y, raise the power of N. Now, in the numerator, we can represent this exponent as a product of exponents. We can rewrite the numerator to be 3Y to the power of N multiplied by 3Y, and this will be divided by 3Y to the power of N. We have a factor of 3Y to the power of N in both the numerator and denominator, so this is going to simplify to be 3Y. Now, what about the N factorial terms? Well, in the numerator, we have N factorial, and in the denominator, we have N + 1 factorial. We can expand the denominator in order to simplify this quotient or this expression, and by expanding the denominator, we will have M + 1 multiplied by N factorial. And this is going to simplify to leave us with 1 divided by N + 1. So, if we apply these simplifications to the limit, we can rewrite the limit to be the limit as N approaches infinity of the quantity 1 divided by M + 1, multiplied by 3 Y. Here, if we were to apply the limit, we will be left with 3 Y divided by infinity, which is going to give us a result of 0. Now, when we apply the ratio test in order to find the radius of convergence and we get a result of zero, then we can assume that the radius is going to be infinite. So we have an infinite radius of convergence, and because the radius of convergence is infinite, then the interval of convergence is going to be all real numbers from negative infinity to positive infinity, and this is going to be the result and the answer to the given 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.

∑ₖ₌₀∞ (2x)ᵏ/k!

Key Concepts

Power Series

Radius of Convergence

Interval of Convergence

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