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: 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...
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: 14 日期: May 25, 2022 节次: 1 In this lecture, we continue the discussion on the root test (for the series with non-negative terms). And we discuss alternating series, there’s a test called Leibniz’s or alternating series test. Alternating series converges in a manner that positive and negative terms...
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,...