$\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 3, 8/29/2022. This page is for Section 1 only.
ACMS 20550: Applied Mathematics Method I
Instructor: Bei Hu, b1hu@nd.edu, Hurley 174A

  1. Special Comparison (I): $\hspace{1em}$ If (a) $b_n>0$, $\sum^\infty b_n$ is convergent, (b) $\lim_{n\to\infty}\frac{a_n}{b_n} = c $ is finite, then $\sum^\infty a_n$ is convergent.

    Special Comparison (II): $\hspace{1em}$ If (a) $d_n>0$, $\sum^\infty d_n$ is divergent, (b) $\lim_{n\to\infty}\frac{a_n}{d_n} = c >0$ , then $\sum^\infty a_n$ is divergent.
  2. Alternating Series:
    For alternating series, if (a) $|a_{n+1}|\le |a_n|$, (b) $\lim_{n\to\infty} a_n = 0$ , then $\sum^\infty a_n$ is convergent.
  3. Conditional convergence: $$\fcolorbox{white}{yellow}{If $\sum^\infty a_n$ is convergent while $\sum^\infty |a_n|$ is divergent, then $\sum^\infty a_n$ is conditionally convergent }$$
  4. Properties.
    1. If $\sum^\infty a_n$ is convergent, so is $\sum^\infty c a_n$

    2. $ \text{If both $\sum^\infty a_n$, $\sum^\infty b_n$ are convergent, so is $\sum^\infty ( a_n+b_n)$ }$

    3. If $\sum^\infty a_n$ is convergent absolutely, so is rearrangement