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\begin{document}

\title{Lesson 16: Integral and Comparison tests for Convergence}



\author[1]{kiersten}%
\affil[1]{Affiliation not available}%


\vspace{-1em}



  
  \date{May 04, 2024}


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\sloppy


\subsection*{\texorpdfstring{\emph{An addition to yesterday's notes . .
.~}}{An addition to yesterday's notes . . .~}}

{\label{930469}}

\section*{Re-Indexing}

{\label{945309}}\par\null

\begin{equation}
\sum_{n=2}^{\infty}(\frac{1}{n-1}) \rightarrow \sum_{j=0}^{\infty}\frac{1}{(j+2)-1} = \sum_{j=0}^{\infty}\frac{1}{j-1}
\end{equation}

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\(\text{let}\ j =\ n -2\)

\(\ \ \ \ \ n=j+2\)
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\subsection*{E}

{\label{631271}}

\subsection*{\(\)}

{\label{631271}}

\begin{equation}
\sum_{n=2}^{\infty}3(\frac{1}{2})^{n} = 3\sum_{j=0}^{\infty}(\frac{1}{2})^{j+2} = 3\sum_{j=0}^{\infty}(\frac{1}{2})^{2}(\frac{1}{2})^{j} = \frac{3}{4}\sum_{j=0}^{\infty}(\frac{1}{2})^{j}
\end{equation}

\section*{Positive Sequences and
Series}

{\label{174412}}

\subsection*{}

{\label{884486}}

\subsection*{\texorpdfstring{\emph{Integral
Test}}{Integral Test}}

{\label{884486}}

\begin{quote}
suppose~\(\text{An}\left(x\right)\leq 0\), then

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\begin{enumerate}
\tightlist
\item
  \(\int_1^{\infty}f(x)dx:C\Rightarrow\sum_{n=1}^{\infty}An:C\)
\item
  \(\int_1^{\infty}f(x)dx:C\Rightarrow\sum_{n=1}^{\infty}An:D\)
\end{enumerate}
\end{quote}
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\par\null

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