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\begin{document}
	
	\mbox{}
	\title{Problem Set 18: Representations of Functions as Power Series}
	
	\maketitle	
	
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%
	%\section{Introduction} \label{sec:intro}
	\noindent {\footnotesize Note: Most of the problems were taken from the textbook} \cite{jS15}.
		
	\begin{problem}
		Find the power series representation for each of the functions below and determine the interval of convergence.
		\begin{enumerate}[label=\alph*), topsep=8pt, itemsep=8pt, partopsep=1ex, parsep=1ex]
			
			\item $\frac 5{1 - 4x^2}$;
			
			\item $\ln(5-x)$;
			
			\item $\frac{x^2}{x^4 + 16}$;
			
			\item $\tan^{-1}(2x)$;
			
			\item $\frac{x-1}{x+2}$;
			
			\item $\frac{x+a}{x^2 + a^2}, \quad a>0$;
			
			\item $\frac{x^2 + x}{(1-x)^3}$;
			
			\item $\ln\big(\frac{1+x}{1-x}\big)$;
			
			\item $x^2 \tan^{-1}(x^3)$.
			
		\end{enumerate}
	\end{problem}
	
	\begin{problem}
		Find the sum of the following series:
		\[
			(a) \ \sum_{n=2}^\infty n(n-1) x^n, \ |x|<1; \quad (b) \ \sum_{n=2}^\infty \frac{n^2 - n}{2^n}; \quad (c) \ \sum_{n=1}^\infty \frac{n^2}{2^n}.
		\]
	\end{problem}
	
%	\begin{problem}
%		Let $p$ and $q$ be real numbers such that $p < q$. Find a power series whose interval of convergence is (a) \ $(p,q)$; (b) \ $(p,q]$; (c) \ $[p,q)$, and (d) \ $[p,q]$.
%	\end{problem}
%	
%	\begin{problem}
%		Is it possible to find a power series whose interval of convergence is $[0,\infty)$? Explain.
%	\end{problem}
	
	\begin{thebibliography}{20}
		
		\bibitem{jS15} J.~Stewart: \emph{Single Variable Calculus} 8th Edition, Cengage Learning, Boston 2015.
		
	\end{thebibliography}
	
\end{document}