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%TCIDATA{<META NAME="Title" CONTENT="Reference/Mathematics/The Integral Test and Estimates of Sums">}
%TCIDATA{Created=Mon Aug 19 14:52:24 1996}
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\section{The Integral Test and Estimates of Sums}

\subsubsection{The Integral Test\label{integral test}}

Suppose $f$ is a continuous, positive, decreasing function on $\left[
1,\infty \right) $ and let $a_{n}=f\left( n\right) $. Then the series $%
\sum\nolimits_{n=1}^{\infty }a_{n}$ is convergent if and only if the
improper integral $\int_{1}^{\infty }f\left( x\right) dx$ is convergent. In
other words,

\begin{enumerate}
\item  if $\int_{1}^{\infty }f\left( x\right) dx$ is convergent, then $%
\sum\nolimits_{n=1}^{\infty }a_{n}$ is convergent.

\item  if $\int_{1}^{\infty }f\left( x\right) dx$ is divergent, then $%
\sum\nolimits_{n=1}^{\infty }a_{n}$ is divergent.\bigskip 
\end{enumerate}

\begin{example}
The $p$\textbf{-series\label{p-series}} $\sum\nolimits_{n=1}^{\infty }\frac{1%
}{n^{p}}$ is convergent if $\dot{p}>1$ and divergent if $p\leq 1$.
\end{example}

\subsubsection{Estimating the Sum of a Series: Remainder Estimate for the
Integral Test\label{remainder estimate for the integral test}}

If $\sum a_{n}$ converges by the Integral Test and $R_{n}=s-s_{n}$, where $s$
is the sum of the series and $s_{n}$ is the sum of the first $n$ terms, then 
\[
\int_{n+1}^{\infty }f\left( x\right) dx\leq R_{n}\leq \int_{n}^{\infty
}f\left( x\right) dx 
\]
The \textbf{remainder} $R_{n}$ is the error made when $s_{n}$ is used as an
approximation to the total sum $s$ of the series.

If you add $s_{n}$ to each side, you get 
\[
s_{n}+\int_{n+1}^{\infty }f\left( x\right) dx\leq s\leq
s_{n}+\int_{n}^{\infty }f\left( x\right) dx 
\]
The inequalities in this equation give a lower and upper bound for $s$. They
provide a more accurate approximation to the sum of the series than the
partial sum $s_{n}$ alone.

\subsection{Related Scientific Notebook Topics}

{\small \hyperref{Integral test}{}{}{../../Help/DM7-7.tex#Integral test}}

\noindent \hrulefill

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