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\noindent
\parbox{2.25in}{\bf Math 3220}
\hfill {\Large \bf Homework \#4} \hfill
\parbox{2.25in}{\bf \hfill Due: Wed., Feb. $27^\mathrm{th}$, 2019}

\vspace{0.2in}

%\noindent {\bf\large Name:} \underline{\qquad\sc\large This is my fabulous name! \qquad}
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\begin{description}
\item[\#1 Comparing Notes] Use the comparison test to prove convergence or divergence of the following series. 

{\it Note:} Use Theorem 2.26 page 127. This is the standard comparison test. Don't use a limit comparison or any other test. {\it Carefully check that all test hypotheses are fulfilled!}

\begin{enumerate}[(a)]
\item $\displaystyle \sum_{k=1}^\infty \dfrac{1}{k+2^k}$ 

\item $\displaystyle \sum_{k=1}^\infty \dfrac{k}{2k^2-1}$ 
\end{enumerate}


\item[\#2 A Series of Questions] State whether the following series either converge absolutely, converge conditionally, or diverge. Use test(s) to prove your assertion. {\it Carefully check that all test hypotheses are fulfilled!} 

\begin{enumerate}[(a)]
\item $\displaystyle \sum_{k=1}^\infty \sin(k)$

\item $\displaystyle \sum_{k=1}^\infty \dfrac{1+\sin^2(k)}{\sqrt{k}}$

\item $\displaystyle \sum_{k=1}^\infty \dfrac{(-k)^3}{k!}$

\item $\displaystyle \sum_{k=1}^\infty \dfrac{(-1)^k}{\sqrt[3]{k}}$ 
\end{enumerate}

\item[\#3 Summing Up] Let $\displaystyle \sum_{k=1}^\infty  a_k$ and $\displaystyle \sum_{k=1}^\infty b_k$ be convergent series.

\begin{enumerate}[(a)]

\item Prove that $\displaystyle \sum_{k=1}^\infty (a_k+b_k)$ converges to $\displaystyle \sum_{k=1}^\infty  a_k + \sum_{k=1}^\infty b_k$.

\item Suppose that $\displaystyle \sum_{k=1}^\infty  c_k$ diverges. Can we conclude that $\displaystyle \sum_{k=1}^\infty (a_k + c_k)$ diverges as well? Prove this or give a counter-example.
\end{enumerate}


\item[RESUBMIT] Type up Homework \#3 Problem \#1 and its solution in \LaTeX. 

\end{description}

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