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

\date{Spring 2006}

\begin{document}

\maketitle

\begin{enumerate}

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

\vfill

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

\vfill

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

\vfill

\clearpage

\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.

\vfill

\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.

\vfill

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


\vfill


\end{enumerate}

 \end{document}