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\title{Math 1b.  Lecture 22 \\
Representations of Functions by Power Series}

\author{T. Judson}

\date{Spring 2006}


\begin{document}

\maketitle


\section{Goals}

\begin{itemize}

\item
To understand and be able to represent functions as power series.

\item
To be able to differentiate and integrate power series to obtain new ways to represent functions as power series.


\end{itemize} 

\section{Representations of Functions by Power Series}


\begin{itemize}

\item 
$\displaystyle \frac{1}{1 - x} = 1 + x + x^2 + x^3 + \cdots$, $|x| < 1$

\item 
$\displaystyle \frac{1}{1 + x} = \frac{1}{1 - (-x)} = 1 - x + x^2 - x^3 + \cdots$, $|x| < 1$

\item 
$\displaystyle \frac{1}{1 + x^2} = 1 - x^2 + x^4 - x^6 + \cdots$, $|x| < 1$

\item 
\begin{eqnarray*}
\frac{x^2}{x+3} 
& = & 
\frac{x^2}{3} \left( \frac{1}{1 - (-x/3)} \right) \\
& = & 
\frac{x^2}{3} \left( 1 - \frac{x}{3} + \frac{x^2}{3^2} - \frac{x^3}{3^3} + \cdots \right) \\
& = &
\frac{x^2}{3}  - \frac{x^3}{3^2} + \frac{x^4}{3^3} - \frac{x^5}{3^4} + \cdots
\end{eqnarray*}


\end{itemize}


\section{Differentiating and Integrating Power Series}

If the power series $\sum c_n (x - a)^n$ has radius of convergence $R > 0$, then the function $f$ defined by
$$
f(x) = c_0 + c_1 (x - a) + c_2 (x - a)^2 + c_3 (x - a)^3 + \cdots = \sum_{n = 0}^\infty c_n (x - a)^n
$$
is term-by-term differentiable and integrable on the interval $(a - R, a+ R)$ and
\begin{itemize}

\item
$\displaystyle f'(x) = c_1  + 2 c_2 (x - a) + 3 c_3 (x - a)^2 + \cdots = \sum_{n = 1}^\infty n c_n (x - a)^{n - 1}$

\item
$\displaystyle \int f(x) \, dx = C + c_0 x + c_1 \frac{(x - a)^2}{2} + c_2 \frac{(x - a)^3}{3} + c_3 \frac{(x - a)^4}{4} + \cdots \\ = C + \sum_{n = 0}^\infty c_n \frac{(x - a)^{n+1}}{n+1}$

\end{itemize}

\section{Examples}

\begin{itemize}

\item
$\displaystyle \arctan x = \int \frac{1}{1 + x^2} \, dx = x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \cdots$

\item
$\ln(x + 1) = \int \frac{1}{1 + x} \, dx = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \cdots$



\end{itemize}




\section{Worksheet Problems}

\begin{center}
\begin{tabular}{|c|c|c|} \hline
Function & Series & Interval of Convergence \\ \hline
$e^x$ &  $\displaystyle 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots$ & $(-\infty, \infty)$ \\  \hline
$\sin x$ &  $\displaystyle x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots$ & $(-\infty, \infty)$ \\ \hline
$\cos x$ & $\displaystyle 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots$ & $(-\infty, \infty)$ \\ \hline
$\displaystyle\frac{1}{1-x}$ & $1 + x + x^2 + x^3 + \cdots$ & $(-1,1)$ \\ \hline
$\displaystyle\frac{1}{1+x}$ & $1 - x + x^2 - x^3 + \cdots$ & $(-1,1)$ \\ \hline
$\displaystyle\frac{1}{1 + x^2}$ & $1 - x^2 + x^4 - x^6 + \cdots$ & $(-1,1)$ \\ \hline
$\arctan x$ & $\displaystyle x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \cdots$ & $[-1, 1]$\\ \hline
$\ln (x + 1)$ & $\displaystyle x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \cdots$ & $(-1,1)$ \\ \hline
\end{tabular}
\end{center}

\begin{enumerate}

\item
Find a power series representation for $\displaystyle f(x) = \frac{x^2}{1+ x}$.

\item
Find a power series representation for $\displaystyle f(x) = \frac{1}{(1+ x)^2}$.

\item
Find a power series representation for $\displaystyle f(x) = \frac{1}{2 + x}$ and determine the interval of convergence.

\item
Find a power series representation for $\displaystyle f(x) = x^2 \cos x$.

\item
Find a power series representation for $\displaystyle f(x) = \cos(x^2)$.

\item
Show that $\displaystyle 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \cdots = \ln 2$.




\end{enumerate}





\section*{References}

\begin{itemize}

\item
\S 8.6 in 
James Stewart.
{\it Single Variable Calculus: Concepts \& Context}, third edition.
Brooks/Cole, Belmont CA, 2005.
ISBN 0--534--41022--7.


\end{itemize}



\section*{Notes}

\subsection*{April 5, 2006}



\end{document}

