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\title{Math 1b.  Lecture 16 \\
The Integral and Comparison Tests; Estimating Sums 
(The Integral Test and $p$-series)}

\author{T. Judson}

\date{Spring 2006}


\begin{document}

\maketitle


\section{Goals}

\begin{itemize}

\item
To understand and be able to apply the integral test to find the convergence of a series.

\item
To understand and be able to tell if a $p$-series converges.


\end{itemize} 

\section{An Example}

The series 
$$
\sum_{n=1}^\infty \frac{1}{n^2}
$$
converges to $\pi^2/6$.  While it requires some additional mathematics to determine this fact, we can, however, determine that the series converges to {\em something},
$$
\sum_{n=1}^\infty \frac{1}{n^2}
=
1+ \sum_{n=2}^\infty \frac{1}{n^2}
\leq
1 + \int_1^\infty \frac{1}{x^2} \, dx.
$$

\section{The Integral Test}

Suppose that $f$ is continuous, positive, and decreasing on $[1, \infty )$ and $a_n = f(n)$.  
\begin{enumerate}

\item
If $\int_1^\infty f(x) \, dx$ is convergent, then $\sum_{n=1}^\infty a_n$ is convergent.

\item
If $\int_1^\infty f(x) \, dx$ is divergent, then $\sum_{n=1}^\infty a_n$ is divergent.

\end{enumerate}
To show this, note that
$$
a_2 + a_3 + \cdots + a_n \leq \int_1^n f(x) \, dx \leq a_1 + a_2 + \cdots + a_{n-1}.
$$
If $\int_1^\infty f(x) \, dx$ is convergent, then
$$
\sum_{k = 2}^n a_k \leq \int_1^n f(x) \, dx \leq \int_1^\infty f(x) \, dx
$$
and
$$
S_n = a_1 + \sum_{k = 2}^n a_k \leq a_1 \int_1^\infty f(x) \, dx = M.
$$
Since $S_{n+1} = S_n + a_{n+1} \geq S_n$ and $S_n \leq M$ for all $n$, the sequence of partial sums is bounded and increasing and must converge.






\section{$p$-Series}

The $p$-series
$$
\sum_{n=1}^\infty \frac{1}{x^p}
$$
converges for $p > 1$ and diverges for $p \leq 1$.

\section{Worksheet Problems}

\begin{enumerate}

\item
Use the integral test to determine the convergence or divergence of
$$
\sum_{n=1}^\infty \frac{1}{\sqrt[4]{n}}.
$$

\item
Use the integral test to determine the convergence or divergence of
$$
\sum_{n=1}^\infty \frac{1}{n^4}.
$$

\item
Use the integral test to determine the convergence or divergence of
$$
\sum_{n=1}^\infty \frac{1}{n^2 + 1}.
$$

\item
If $f(k) = a_k$, rank the values $\int_1^n f(x) \, dx$, $\sum_{k=1}^{n-1} a_k$, and $\sum_{k=2}^{n}a_k$ in increasing order.

\item
If $f(k) = a_k$, rank the values $\int_n^\infty f(x) \, dx$, $\sum_{k=n+1}^\infty a_k$, and $\int_{n+1}^\infty f(x) \, dx$ in increasing order.

\item
For what values of $p$ is the series
$$
\sum_{n=2}^\infty \frac{1}{n (\ln n)^p}
$$
convergent?



\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 17, 2006}



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