前言 Zotero 是文献管理利器。Notion 是第二大脑。那么二者能否结合起来，最大程度地提高生产效率呢？周末我突然想到这个问题，网上搜了一下，还真有现...
在 Emacs + AucTeX 中编译主 LaTeX 文件
背景 尽管使用 Emacs + AucTeX 编译 TeX 文件已经十多年了，但非常惭愧的是我一直是用最最基本的功能，比如 C-c C-c 编译 LaTeX 文件，C-c C-l 查看编译错误信息等等，从来没有...
W16-S2 Applications of Power Series
周次: 16 日期: June 10, 2022 节次: 2 In this lecture, various applications for Taylor series are demonstrated. And we show how to generate a Fourier series from $f(x)$ in the end of the lecture. § 11.10 Applications of Power Series The Binomial Series for Powers and Roots The Taylor series generates by $f(x) = (1+x)^a$, when $a$ is constant, is called the binomial...
W16-Extra1 Convergence of Taylor Series; Error Estimates
周次: 16 日期: June 9, 2022 节次: Extra1 In this lecture, we estimate the error of Taylor polynomial to the generating function. The result provides a criteria for the convergence of Taylor series. A lot of common functions can be expanded to Taylor series according to this result. And without the proof, we will see that more common functions can be represented by power...
W16-S1 Taylor Series and Maclaurin Series
周次: 16 日期: June 8, 2022 节次: 1 In this lecture, we show that the convergent power series represents a function $f(x)$. The coefficient of the power series can be presented in turn by the function. This opens a new page for the application of power series, for we can associate power series to any infinitely differentiable functions. To see the power series derived...
W15-S1 Power Series (II)
周次: 15 日期: June 1, 2022 节次: 1 In this lecture, we will continue the discussions on the power series. A radius of convergence of a power series is emerging from the ratio test for the absolute convergence. We will discuss the basic operations, like differentiation and integrations on a power series, and the multiplication of two power series as well. When a power...