$\newcommand{\dis}{\displaystyle} \newcommand{\m}{\hspace{1em}} \newcommand{\mm}{\hspace{2em}} \newcommand{\x}{\vspace*{1ex}} \newcommand{\xx}{\vspace*{2ex}} \let\limm\lim \renewcommand{\lim}{\dis\limm} \let\fracc\frac \renewcommand{\frac}{\dis\fracc} \let\summ\sum \renewcommand{\sum}{\dis\summ} \let\intt\int \renewcommand{\int}{\dis\intt} $

Lecture 5, 9/2/2022. This page is for Section 1 only.
ACMS 20550: Applied Mathematics Method I
Instructor: Bei Hu, b1hu@nd.edu, Hurley 174A

  1. Taylor expansion:
    $f(x) = \sum_{n=0}^{\infty} a_n (x-a)^n, \mm a_0= f(a), \m a_1 = f'(a), \m a_2 = \frac1{2!} f''(a), \m \cdots, \m a_n = \frac1{n!} f^{(n)}(a).$

    $\fcolorbox{white}{yellow}{When $a=0$, the Taylor series is also called the Maclaurin Series.}$
  2. Some Basic Taylor (Maclaurin) Series:
    1. $\sin x = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!} = x- \frac{x^3}{3!} + \frac{x^5}{5!}-\frac{x^7}{7!}+ \cdots, \m |x|<\infty$
    2. $\cos x = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!} = 1- \frac{x^2}{2!} + \frac{x^4}{4!}-\frac{x^6}{6!}+ \cdots,\m |x|<\infty $
    3. $e^x = \sum_{n=0}^{\infty} \frac{ x^{n}}{n!} = 1+x+ \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \cdots, \m |x|<\infty $
    4. $\ln(1+x) = \sum_{n=1}^{\infty} \frac{(-1)^{n+1} x^{n}}{n} = x- \frac{x^2}{2 } + \frac{x^3}{3 } -\frac{x^4}{4 } + \cdots, \m -1< x \le 1 $
    5. $(1+x)^p = \sum_{n=0}^{\infty} \pmatrix{p\cr n} x^{n} = 1+p x+ \frac{p(p-1)}{2! }x^2 + \frac{p(p-1)(p-2)}{3! } x^3 + \cdots, \m |x|<1 $
    6. Don't forget the formula from Lecture 1: $\mm \frac1{1-x} = \sum_{n=0}^{\infty} x^{n} = 1 + x + x^2 + x^3 + x^4 + \cdots, \m |x|<1 $