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\title{\vspace*{-1.5in}\textbf{\S8.5: Power Series}\\\normalsize (Created by Faan Tone Liu)}
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\maketitle
%\underline{Math 1300-400: Quiz 1}
\lhead{Math 2300-007: Calculus II}
\rhead{March 16, 2018}
\chead{\S8.5: Power Series}
\vspace{-.8in}
\noindent
\textbf{Key Points:}
\begin{enumerate}[\textbf{A.}]
\item What is a \textbf{power series}?
\begin{itemize}
\item First Perspective: Inspired by polynomials, we create an ``infinite-degree polynomial.'' For example:\vskip2cm

\item Second Perspective: Put a $x^n$ as part of a series. For example:\vskip2cm

\item Third Perspective: A power series is a function where $x$ is the input and the output is a series. For example:\vskip2cm
\end{itemize}

\item Basic questions:
\begin{itemize}
\item For what $x$-values dos the power series converge? To answer this question, use the \underline{\phantom{ratio test ratio test}}. The result is an interval called the \textbf{interval of convergence}. Important: Check the endpoints separately.
\item To what value does the series converge? 
\end{itemize}
\end{enumerate}
\noindent
\textbf{Examples:}\vskip12pt
\begin{enumerate}
\item Consider the series $\displaystyle 1 + x + x^2 + \cdots + x^n + \cdots.$ For which values of $x$ does the series converge?\newpage
\item Find the interval of convergence of the series $\displaystyle 1 + x + \frac{x^2}{2} + \frac{x^3}{3!} + \cdots.$\vfill
\item Find the interval of convergence of the series $\displaystyle 1 - \frac{(x-3)}{2}+ \frac{(x-3)^2}{4} - \frac{(x-3)^3}{8} + \cdots.$\vfill
\item Find the interval of convergence of the series $\displaystyle \sum_{n=0}^\infty n!x^n$.\vfill
\item Find the interval of convergence of the series $\displaystyle \sum_{n=0}^\infty \frac{(2x+1)^n}{n3^n}$.\vfill
\end{enumerate}
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