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\title{Math 1b. Series---Other Convergence Tests 
(Alternating Series, Absolute and Conditional Convergence)}

\date{Spring 2006}

\begin{document}

\maketitle

\begin{enumerate}

\item 
Determine the convergence of $\displaystyle \sum_{n=1}^\infty \frac{(-1)^{n+1}}{\sqrt{n}}$.

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\item 
Determine the convergence of $\displaystyle \sum_{n=1}^\infty \frac{(-1)^{n+1}}{n^2 + 2^n}$.

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\item 
Suppose that $a_n \geq 0$ for all $n \geq 1$.  Is is possible for $\sum_{n=1}^\infty a_n$ to converge conditionally?

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\clearpage

\item 
For what values of $p$ does the series $\displaystyle \sum_{n=2}^\infty (-1)^{n+1}\frac{\ln k}{n^p}$ converge absolutely?

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\item 
For what values of $p$ does the series $\displaystyle \sum_{n=2}^\infty (-1)^{n+1}\frac{\ln k}{n^p}$ converge conditionally?

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\item 
Determine the convergence of $\displaystyle 1 - \frac{1}{2^3} + \frac{1}{3^2} - \frac{1}{4^3} + \frac{1}{5^2} - \frac{1}{6^3} +  \frac{1}{7^2} - \frac{1}{8^3} +\cdots $.

\vfill


\end{enumerate}


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