“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

Infinite radius. [MUSIC] It can certainly happen at the radius of convergence is infinite. Well let's consider this power series. The sum n grows from 0 to infinity of x to the n divided By n factorial. Let's try the ratio test. So here we go, the limit n goes to infinity of the n plus first term. So x to the n+1 over (n+1)! divided by this the nth term. So let's display here Xn over n factorial, and absolute value of that because I'm checking for absolute convergence with the ratio test. I can simplify this a bit. This is a fraction with fractions in the numerator and denominator. This is the limit n goes to infinity of Xn + 1 times n factorial in the denominator of the denominator, put that in the numerator. Divided by x to the n times n plus 1 factorial. So, I've just got a fraction, but I can simplify this a bit, too. This is the limit n goes to infinity. We've got x to the n plus 1 over x to the n, just leaves you with x in the numerator. And I've got n factorial in the numerator and n plus 1 factorial in the denominator. Well, n plus 1 factorial kills everything here, right? N plus 1 factorial is 1 times 2, all the way through n plus 1. And that contains all the terms and n factorials. So what I'm left with is just an n plus 1 in the denominator. Now, what is this limit? X is just some fixed quantity. It doesn't depend on n. But what's the limit then of sum number x divided by n + 1, n going to infinity. Well this limit is 0. It doesn't matter what x is. If you take some fixed number divided by a very large quantity, you could make this equals to 0 as you like. So this limit is 0, 0 is less than 1. And that means by the ratio test this series converges regardless of what x is.

So what's the radius of convergence? Series converges for all values of x. And that means the radius of convergence is infinity. And in the not-too-distant future, we're going to see a very surprising result. We're eventually going to see that this series is in fact a complicated way of writing down a function we already know. This is just a complicated way of writing down the function e to the x. Meaning that if you plug in a specific values for x say, x equals negative 1. You get that the sum n goes 0 to infinity of -1 to the n over n factorial is equal e to the -1 is equal to 1 over e. And by choosing value of x, by plugging in different specific values of x. We can generate a ton of really neat series. Here's another example. Just the fact that the sum N goes from zero to infinity of ten to the N over in factorial is well, according to this, that's just E to the 10th power. And that is really cool. But let me warn you and share a bit of the philosophy of power series with you. Yes. By plugging in specific values of X, you can generate a ton of interesting examples but power series aren't just a way of generating a bunch of series in isolation. Part of the joy of power series comes by thinking of power series not as a mechanism for generating a bunch of discreet examples. But as a way of collecting together a whole bunch of interesting series that depend on a parameter x. [SOUND]

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