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\title{Math 1b.  Lecture 23 \\
Taylor and Maclaurin Series I}

\author{T. Judson}

\date{Spring 2006}


\begin{document}

\maketitle


\section{Goals}

\begin{itemize}

\item
To understand and be able to compute Taylor and Maclaurin series.

\item
To understand and be able to find the Taylor polynomial of a function.




\end{itemize} 

\section{Does Every Function Have a Power Series Representation?}

This is certainly not the case since a function must be infinitely differentiable.  Thus, $f(x) = |x|$ has no power series representation at $x = 0$.  If $f$ does have a power series representation, then the following must be true.

\bigskip

\fbox{\parbox{4.5in}{
If a function $f$ has a power series representation at $x = a$, 
$$
f(x) = \sum_{n = 0}^\infty c_n (x - a)^n \qquad |x - a| < R,
$$
then the coefficients of the power series are given by
$$
c_n = \frac{f^{(n)}(a)}{n!}.
$$
}}

\bigskip

\section{Taylor and  Maclaurin Series}



\begin{itemize}

\item
A {\em Taylor Series\/} for $f$ centered at $x = a$ is given by
$$
f(x) =  \sum_{n = 0}^\infty \frac{f^{(n)}(a)}{n!} (x - a)^n \\
= f(a) + f'(a)(x - a) + \frac{f''(a)}{2!}(x - a)^2 + \frac{f'''(a)}{3!}(x - a)^3 + \cdots.
$$

\item
A Taylor series centered at $x = 0$ is called a {\em Maclaurin series},
$$
f(x) =  \sum_{n = 0}^\infty \frac{f^{(n)}(0)}{n!} x^n \\
= f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3 + \cdots
$$

\end{itemize}
Thus, for a given a function $f$ that is infinitely differentiable, we can compute its Taylor series.  The question of whether or not the series converges remains.

\section{Taylor Polynomials}

The $n$th degree {\em Taylor Polynomial\/} of a function $f$ is
$$
T_n = f(a) + f'(a)(x - a) + \frac{f''(a)}{2!}(x - a)^2 + \frac{f'''(a)}{3!}(x - a)^3 + \cdots + \frac{f^{(n)}(a)}{n!} (x - a)^n.
$$
The Taylor polynomials approximate $f$ near $a$.  In other words, the Taylor polynomials are the partial sums of a Taylor series.

\section{Worksheet Problems}

\begin{enumerate}


\item
Find the Taylor polynomial for $f(x) = \cos x$, where $n = 4$ and $a = 0$

\item
Find the Taylor polynomial for $f(x) = \cos x$, where $n = 4$ and $a = \pi/6$

\item
Find the Taylor series for $f(x) = \cos x$, where $a = 0$.


\item
Find the Taylor series for $f(x) = x^3 - 3x^2 + 2$, where $a = 2$.

\item
Find the Taylor series for $f(x) = \ln x$, where $a = 1$.

\end{enumerate}





\section*{References}

\begin{itemize}

\item
\S 8.7 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 7, 2006}



\end{document}

