“微积分二：数列与级数”将介绍数列、无穷级数、收敛判别法和泰勒级数。本课程不仅仅满足于得到答案，而且要做到知其然，并知其所以然。

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From the course by The Ohio State University

微积分二: 数列与级数 (中文版)

45 ratings

The Ohio State University

45 ratings

“微积分二：数列与级数”将介绍数列、无穷级数、收敛判别法和泰勒级数。本课程不仅仅满足于得到答案，而且要做到知其然，并知其所以然。

From the lesson

幂级数

在第五个模块中，我们学习幂级数。截至目前为止，我们一次讲解了一种级数；对于幂级数，我们将讲解整个系列取决于参数 x 的级数。它们类似于多项式，因此易于处理。而且，我们关注的许多函数，如 e^x，也可表示为幂级数，因此幂级数将轻松的多项式环境带入棘手的函数域，如 e^x。

- Jim Fowler, PhDProfessor

Mathematics

Differentiate. [MUSIC] Differentiating polynomials, it's not so bad. I mean, for example, if I wanted to differentiate some polynomial, maybe (2x- 4x cubed + 3x to the 10th), say. All I've gotta do is just remember my rules for differentiating. I differentiate these sums and differences by differentiating each term. And then I can differentiate a power just by bringing down the power and subtracting 1, right? So the derivative of 2x is just 2. The derivative of 4x cubed is 12x squared. The derivative of 3x to the 10th is 30x to the 9th. Turns out it's not too much worse to differentiate a power series. Well here's how you do it, it's a theorem. Suppose I've got some power series and I'm calling that f(x). And R is the radius of convergence of this power series. So for any x between -R and R, this function f(x) is defined to be the value of this thing, convergent power series. Now here's the theorem, then. The derivative of this function f is this. The sum n goes from 1 to infinity of n times an times x to the n- 1. And if that looks mysterious, where is that coming from? Well that's just the derivative of an times x to the n. So this is telling you that if you want to differentiate a function which is given to you as a power series, well then the derivative is just the sum of the derivatives of the terms of the power series. You can differentiate term by term. This new power series has the same radius of convergence as the old power series. And this power series, for any value of x between -R and R, is equal to the derivative of this function which is given to you as this power series. At this point, you might be wondering why I'm even calling this a theorem. I mean, what's the big deal? And you might be thinking or remembering that the derivative of the sum is the sum of the derivatives. So what's the big deal? I mean, isn't this something that we already know?

The situation here, that the derivative of a power series is the power series of the derivatives, that's actually way more subtle. Well the issue is that that's not really what we're talking about. It is true that the derivative of a sum is just the sum of the derivatives. But what I'm asking here is whether the derivative of a series is the series of the derivatives. That's really something more complicated, right? What's the definition of the series? Well it's the derivative of the limit of the partial sums. And I'm wondering, is that the limit of the sum of the derivatives? And although we do have a theorem that the derivative of a sum is the sum of the derivatives, we don't have a theorem. And it's not true in general that the derivative of a limit is the limit of the derivatives. So it's a big deal. The fact that you can differentiate a power series term by term, it's a theorem. I mean, that's really something that's not obvious. We're not going to prove this theorem. But we are going to make use of it. And I think you'll find that it's extraordinarily helpful.

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