Interval of convergence for derivative and integral (video) | Khan Academy

Video transcript- [Instructor] Times in ourdealings with powers series. We might wanna take the derivative, or we might want to integrate them. And in general, we cando this term by term. What do I mean by that? Well, that means that the derivative of f, f prime of x, is justgonna be the derivative of each of these terms. So that's gonna be the sumfrom n equals one to infinity. And let's see, thederivative of x to the n is n times x to the n minus one. So I could write this as ntimes x to the n minus one all of that over n. And these ns will cancel out, so this is just going to be, this is just going tobe x to the n minus one. So this is taking thederivative with respect to x. Similarly, we could integrate, we could integrate and we could evaluate, we could evaluate theintegral of f of x dx, and this is going to beequal to some constant plus, if we integrate this term by term. And so this is goingto be equal to the sum from n equals one to infinity. And let's see, we increment the expo...