$\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} $

Lecture 1, 8/24/2022. This page is for Section 1 only.
ACMS 20550: Applied Mathematics Method I
Instructor: Bei Hu, b1hu@nd.edu, Hurley 174A

  1. Geometric Serise
    $ \sum_{n=0}^\infty a r^{n} = \frac{a}{1-r} = \frac{ \text{leading term}}{1-\text{ratio}} , \hspace{1em} |r|<1.$
  2. limit of rational type
    If order of $P$ = order of $Q$, $\hspace{1em} \lim_{n\to\infty} \frac{P(n)}{Q(n)} = \frac{\text{highest order coefficient of }P(n)}{\text{highest order coefficient of }Q(n)} $
    $$e.g., \hspace{2em} \lim_{n\to\infty}\frac{(5n+1)^3+\sqrt{1+16n^6}}{9+7n+5n^3} = \frac{5^3+\sqrt{16}}{5} = \frac{129}{5}, \hspace{1em} \text{order} = 3 $$
  3. L'Hôpital's rule is handy sometimes.
  4. Convergence and divergence $$\fcolorbox{white}{yellow}{ Partial Sum: $\hspace{1em} S_n = a_1+a_2+\cdots +a_n = \sum_{j=1}^n a_j $} \hspace{1em} \fcolorbox{white}{yellow}{ Series: $\hspace{1em} S = \lim_{n\to\infty} S_n = \sum_{j=1}^\infty a_j $ } \hspace{1em} \fcolorbox{white}{yellow}{ Remainder: $\hspace{1em} R_n= S-S_n \rule[-3ex]{0pt}{6.5ex} $} $$
  5. Preliminary test (Divergence test):
    If $\hspace{1em} \lim_{n\to\infty} a_n \neq 0$, then $ \sum_{n=1}^\infty a_n $ is divergent