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\title{Math 1b. Series---Definitions and Properties}

\date{Spring 2006}

\begin{document}

\maketitle



\begin{enumerate}

\item
Determine the convergence or divergence of each of the following series.  If the series converges, try to find the sum.
\begin{enumerate}

\item 
$\displaystyle \sum_{n = 1}^\infty \tan n$

\vfill

\item 
$\displaystyle \sum_{n = 1}^\infty \frac{n+1}{2n - 3}$

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\item 
$\displaystyle \sum_{n = 1}^\infty  \frac{2}{n^2 + 4n + 3}$

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\item 
$\displaystyle \sum_{n = 1}^\infty \ln\left(  \frac{n}{n+1} \right)$


\vfill




\end{enumerate}

\clearpage

\item
Let $\{ a_n\}$ be an increasing sequence such that $a_1 >0$ and $a_n \leq 100$ for all $n \geq 1$.
\begin{enumerate}

\item
Does $\displaystyle \lim_{n \to \infty} a_n$ converge?

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\item
Does $\displaystyle \sum_{n=1}^\infty a_n$ converge?

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\end{enumerate}

\item
Suppose that the partial sums of $\sum_{n=1}^\infty a_n$ satisfy the inequality
$$
\frac{6 \ln n}{\ln(n^2 + 1)} < S_n < 3 + ne^{-n}
$$
for all $n > 100$.
\begin{enumerate}

\item
Does $\displaystyle \sum_{n=1}^{\infty} a_n$ converge?  If so, what is the limit? 

\vfill

\item
Can anything be said about   $\displaystyle \lim_{n \to \infty} a_n$?


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\end{enumerate}



\end{enumerate}

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