$\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 2, 8/26/2022. This page is for Section 1 only.
ACMS 20550: Applied Mathematics Method I
Instructor: Bei Hu, b1hu@nd.edu, Hurley 174A

  1. More Tests:
    1. Comparison test: $\hspace{1em}$ If (a) $ \sum_{n=1}^\infty m_n$ is convergent, (b) $|a_n|\le m_n$, then $ \sum_{n=1}^\infty a_n $ is convergent absolutely

      $\fcolorbox{white}{yellow}{ Convergent absolutely : convergnet after taking absolute value } $
      Comparison test: $\hspace{1em}$ If (a) $ \sum_{n=1}^\infty d_n$ is divergent, (b) $a_n\ge d_n \ge 0$, then $ \sum_{n=1}^\infty a_n $ is divergent


    2. $ \text{Integral test: $\hspace{1em}$ If $0< a_{n+1}\le a_n$, then $\sum^\infty a_n$ is convergent if and only if $\int^\infty a_n dn < \infty$}$
    3. Ratio test: $\hspace{1em}$ Let $\rho = \lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|.$ Then $\hspace{1em} \left\{ \begin{array}{c} \text{If $\rho<1$, then $\sum^\infty a_n$ is convergent }\cr \text{If $\rho=1$, no conclusion (need further work) }\cr \text{If $\rho>1$, then $\sum^\infty a_n$ is divergent} \end{array}\right. $