“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

交错级数

在第四个模块中，我们讲解绝对和条件收敛、交错级数和交错级数审敛法，以及极限比较审敛法。简而言之，此模块分析含有一些负项和一些正项的级数的收敛性。截至目前为止，我们已经分析了含有非负项的级数；如果项非负，确定敛散性会更为简单，因此在本模块中，分析同时含有负项和正项的级数，肯定会带来一些新的难题。从某种意义上，此模块是“它是否收敛？”的终结。在最后两个模块中，我们将讲解幂级数和泰勒级数。这最后两个课题将让我们离开仅仅是敛散性的问题，因此如果你渴望新知识，请继续学习！

- Jim Fowler, PhDProfessor

Mathematics

Alternating series. [SOUND] What is an alternating series? For example, this series, the sum n equals 1 to infinity of -1 to the n plus 1 over n is an example of an alternating series. What does that even mean? Let me write down some of the terms I think will be a little bit clearer. So if I plug in n = 1, I get -1 squared which is 1 over 1. So 1 over 1. When I plug in n equals 2, I get -1 cubed over 2, that's -1 over 2, so that's minus a half. When I plug in n equals 3, that's negative 1 to the fourth, over 3, that's 1 over 3, that's plus a third. When I plug in n equals 4, I get negative 1 to the fifth, which is negative 1 over 4, so minus a fourth. When I plug in n equals five, I get negative one to the sixth, over five, that's 1 over 5, so plus a fifth. When I plug in n equals six, I get negative one to the seventh, over six, so that's just negative 1 Over 6, so minus a 6th. And then it's going to keep on going like that. But I'm flip-flopping between these two colors. I'm alternating signs, right? I'm adding, subtracting, adding, subtracting, adding, subtracting. It's an alternating series. So that's what I mean by alternating. Well, here's a precise definition. The series, sum a sub n, n goes from 1 to infinity, is called an alternating series. If a sub n = (-1) to the nth power * b sub n.

And all the b sub n are the same sign. So maybe the b sub n sequence is they're all positive. And then a sub n has this term here, times a positive sequence. Well, this term is what makes the signs, the S-I-G-Ns, flip-flop back and forth. Well here's a bit of a warning. Not every series which has both positive and negative terms is an alternating series. For example, this series, the sum n goes from 1 to infinity of -1 to the n times sine n over n. This is not an alternating series. I mean, yes, it's got the -1 to the n here. But the sine of n also introduces its own quite complicated pattern of positive and negative terms. So this is a series, some of the terms are positive, some of the terms are negative, but it's not alternating, right? This is not an alternating series. A couple of other examples. Sum n goes from 1 to infinity, say -1 to the n times n plus 1 over 2. So that's all in the exponent there, n times n plus 1 over 2, say all over n. This is also not an alternating series. And it's still got a minus 1 there to raise to some power but the power's a little bit more complicated. N times n plus 1 over 2, at terms or some of them maybe positive, some of them maybe negative but they're not flip flopping in sign, in S-I-G-N back and forth. In contrast here's an example that is an altering series, the sum n goes from one to infinity say of -1 to the n times sine squared n over n. Sine squared n, this turns out to always be positive, right? The n is positive here and -1 to the n. This is the only thing that's affecting the S-I-G-N of this term. This does indeed in flip flop back and forth between negative positive, negative positive, negative positive. This is an alternating series. [SOUND]

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