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...
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...
周次: 14 日期: May 27, 2022 节次: 2 In this lecture, we will continue the discussion of absolute and conditional convergence, give the absolute convergence test to show any absolutely convergent series itself must converge. We also look at the difference between absolute and conditional convergences by showing that an absolute convergent series remains the same even after the rearrangement of the terms. However,...
周次: 13 日期: May 18, 2022 节次: 1 In this lecture, we develop some convergence tests for the series with non-negative terms. These are the Integral Test and Comparison Tests. Both tests are based on the Monotonic Bounded Theorem for partial sum sequences. Before developing the tests, we will recap the previous section first. I. Recap: § 11.2 Series Basic Problems for Series Primarily,...
周次: 12 日期: May 13, 2022 节次: 2 In this lecture, we will talk about series, we introduce the sigma notation, and utilize limit to define the sum of a series. Primarily, there are two types of series whose sums can be found directly, by definition. Those are called Geometric Series and Telescoping Series. To study a divergent series, we introduce the $n$th term...
周次: 12 备注: Attendance check 日期: May 11, 2022 节次: 1 In this lecture, we continue the discussion of infinite series. We introduce the L’Hôpital’s Rule to the limit finding problems, and conclude this section by an important type of convergent sequences, called bounded monotonic sequences. Then we will talk about series,...