“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

Does it converge?

There's a somewhat standard

process that you can use to go about checking the convergence of a series.

You've got a series, the sum n goes from 1 to infinity a sub n, and

you want to know, does it converge?

I'd recommend applying the limit test first.

Because if you calculate the limit of a sub n as n goes to infinity and

that's not zero, then you know the series diverges and you're done.

Then, I would try to check absolute convergence.

Right.

So if this limit is zero, then you don't know All right.

You could try to investigate absolute convergence and you could do that

using any of the tests that we have for series, returns, and nonnegative,

the root test, the ratio test, the limit comparison test, what have you.

And if it converges absolutely, you're done.

But if it doesn't converge absolutely Well, in that case,

you're back to just looking at the sum of the a sub n's again.

If your series has some positive and some negative terms and it doesn't converge

absolutely, well, you'd better hope that it's an alternating series.

Because if this is an alternating series,

then at least you have the alternating series test at your disposal.

If it's not an alternating series, well you could try writing down the sequence

of partial sums, and try to evaluate the limit just with your bare hands.

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