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\title{Math 1b.  Lecture 17 \\
The Integral and Comparison Tests; Estimating Sums 
(Comparison Tests)}

\author{T. Judson}

\date{Spring 2006}


\begin{document}

\maketitle


\section{Goals\protect\footnote{We will not require that students know the Limit Comparison Test}}

\begin{itemize}

\item
To understand and be able to apply the Comparison Test to determine the convergence or divergence of a series.

\item
To understand and be able to the remainder of a series using the Integral Test.


\end{itemize} 

\section{The Comparison Test}

Let $\sum a_n$ and $\sum b_n$ be series with positive terms.
\begin{itemize}

\item
If $\sum b_n$ is convergent and $a_n \leq b_n$ for all $n$, then $\sum a_n$ also converges.

\item 
If $\sum b_n$ is divergent and $a_n \geq b_n$ for all $n$, then $\sum a_n$ also diverges.

\end{itemize}

For example, suppose that we wish to determine the convergence of 
$$
\sum_{n=1}^\infty \frac{1}{n^2 + 2n + 1 + \sin^2 n}.
$$
Since
$$
\frac{1}{n^2 + 2n + 1 + \sin^2 n} \leq \frac{1}{n^2}
$$
for $n = 1, 2, \ldots$ and the series $\sum_{n=1}^\infty 1/n^2$ converges, the original series must also converge.



\section{Computing the Remainder of a Series}

\begin{itemize}

\item
Suppose that we wish to estimate $\sum_{n=1}^\infty 1/n^5$ to three decimal places.

\item
The remainder of a convergent series $S = \sum_{n=1}^\infty a_n$ with partial sums $S_N = \sum_{n=1}^N a_n$ is
$$
R_N = S - S_n = a_{N+1} + a_{N+2} + \cdots.
$$
We know that
$$
\int_{N+1}^\infty f(x) \, dx, \leq R_N \leq \int_N^\infty f(x) \, dx,
$$
where $f(k) = a_k$.

\item
Solving the inequality 
$$
R_N \leq \int_N^\infty \frac{1}{x^5} \, dx = \frac{1}{4 N^4} \leq 0.0005,
$$
we can choose $N = 5$,

\end{itemize}



\section{Worksheet Problems}

\begin{enumerate}

\item 
Determine the convergence of the series
$$
\sum_{n=1}^\infty \frac{n^2 - 1}{3n^4 + 1}.
$$

\item 
Determine the convergence of the series
$$
\sum_{n=1}^\infty \frac{n^2 - 1}{3n^4 + 1}.
$$

\item 
Determine the convergence of the series
$$
\sum_{n=1}^\infty \frac{4 + 3^n}{2^n}.
$$

\item 
Determine the convergence of the series
$$
\sum_{n=1}^\infty \frac{1 + \sin n}{10^n}.
$$

\item 
Estimate the series
$$
\sum_{n=1}^\infty \frac{1}{n^{3/2}}
$$
to within 0.01.

\item
If we estimate $\sum_{n=1}^\infty 1/n^{3/2}$ by
$$
\sum_{n=1}^{10} \frac{1}{n^{3/2}},
$$
how accurate is our estimate?




\end{enumerate}





\section*{References}

\begin{itemize}

\item
\S 8.3 in 
James Stewart.
{\it Single Variable Calculus: Concepts \& Context}, third edition.
Brooks/Cole, Belmont CA, 2005.
ISBN 0--534--41022--7.


\end{itemize}



\section*{Notes}

\subsection*{March 20, 2006}



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