$\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 4, 8/31/2022. This page is for Section 1 only.
ACMS 20550: Applied Mathematics Method I
Instructor: Bei Hu, b1hu@nd.edu, Hurley 174A

  1. Power Series:
    $ \fcolorbox{white}{yellow}{Power Series (expansion at center $x=0$): $\sum_{n=0}^\infty a_n x^n = a_0 + a_1 x + a_2 x^2 + a_3 x^3 + \cdots$ }$
    $ \fcolorbox{white}{yellow}{or Power Series (expansion at center $x=a$): $\sum_{n=0}^\infty a_n (x-a)^n = a_0 + a_1 (x-a) + a_2 (x-a)^2 + a_3 (x-a)^3 + \cdots$ } $
    $\fcolorbox{white}{yellow}{ Radius of convergence $R$: the series is convergent for $|x-a|< R$. }$
    $\frac1R = \lim_{n\to\infty} \frac{|a_{n+1}|}{|a_n|} \fcolorbox{white}{yellow}{Caution: the formula fails if the limit does not exist}$
  2. Some big theorems:
    1. Within the common interval of convergernce, two power series can be (a) added, (b) subtracted,
      (c) multiplied, (d) divided (provided that the denominator is not zero)
    2. Within the radius of convergence, a power series can be term by term (a) differentiated, (b) integrated.
    3. Composition of two power series is another power series with $\fcolorbox{white}{yellow}{new radius of convergence.}$