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 )
- $e^x = 1 + \frac{x}{1!} + \frac{x^2}{2!} + \frac{x^3}{3!}+\cdots$
- $\cos x = 1- \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!}+\cdots$
- $\sin x = x -\frac{x^3}{3!} + \frac{x^5}{5!} -\frac{x^7}{7!}+\cdots$
- $(1+x)^p = 1 + px + \frac{p(p-1)}{2!}x^2 + \frac{p(p-1)(p-2)}{3!}x^3 + \cdots$
Explain the similarity between the exponential and trigonometrical series using complex numbers
Justify the convergence of the exponential series to the exponential function