Taylor series

Learning objectives

  • Define linear and quadratic taylor polynomials
  • Work with the 4 most important Taylor series: exponential, cosine, sine, and binomial (which generalizes the geometric series for (1+x)^{-1})
    • e^x = 1 + \frac{x}{1!} + \frac{x^2}{2!} + \frac{x^3}{3!}+\cdots
    • \cos x = 1- \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!}+\cdots
    • \sin x = x -\frac{x^3}{3!} + \frac{x^5}{5!} -\frac{x^7}{7!}+\cdots
    • (1+x)^p = 1 + px + \frac{p(p-1)}{2!}x^2 + \frac{p(p-1)(p-2)}{3!}x^3 + \cdots
  • Explain the similarity between the exponential and trigonometrical series using complex numbers
  • Justify the convergence of the exponential series to the exponential function

Before class

  • Read Hughes-Hallett 10.1-10.3
  • Ponder Hughes-Hallett problems 10.1:4,7,12; 10.2:34,41,43; 10.3:1,2,3,8,21.

In class