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\title{Math 1b.  Lecture 24 \\
Taylor and Maclaurin Series II}

\author{T. Judson}

\date{Spring 2006}


\begin{document}

\maketitle


\section{Goals}

\begin{itemize}



\item
To understand the convergence of a Taylor series.


\item
To understand and be able to compute Taylor's Inequality.


\end{itemize} 

\section{The Remainder of a Taylor Series}

If $f$ has a Taylor series and
$$
T_n = f(a) + f'(a)(x - a) + \frac{f''(a)}{2!}(x - a)^2 + \frac{f'''(a)}{3!}(x - a)^3 + \cdots + \frac{f^{(n)}(a)}{n!} (x - a)^n
$$
is the $n$th degree {\em Taylor Polynomial\/} of a function $f$, then
$$
f(x) = \lim_{n \to \infty} T_n.
$$
Define the {\em remainder\/} of a Taylor series to be
$$
R_n(x) = f(x) - T_n(x).
$$
If $R$ is the radius of convergence of the series, then
$$
\lim_{n \to \infty} R_n = 0
$$
for $|x - a| < R$.

\section{Taylor's Inequality}

If $|f^{(n + 1)}| \leq M$ between $a$ and $x$, then the remainder of the Taylor series satisfies the inequality
$$
|R_n(x)| \leq \frac{M}{(n+1)!} |x - a|^{n+1}.
$$

\section{Taylor Series of Some Familiar Functions}

\begin{itemize}

\item
$\displaystyle e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots$

\item
$\displaystyle \sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots$

\item
$\displaystyle \cos x = 1 -  \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots$

\end{itemize}

\section{Multiplication and Division of Power Series}

Power series can be multiplied and divided just like polynomials.  For example,
$$
e^x \sin x = x + x^2 + \frac{x^3}{3} + \cdots.
$$






\section{Worksheet Problems}

\begin{enumerate}


\item
Let
$$
f(x) = \sum_{n = 0}^\infty \frac{(-1)^n x^{2n}}{(2n + 1)!} = 1 - \frac{x^2}{3!} + \frac{x^4}{5!} - \frac{x^6}{7!} + \cdots 
$$
for all real numbers $x$.
\begin{enumerate}

\item
Find $f'(0)$ and $f''(0)$.  Determine whether $f$ has a local minimum, a local maximum, or neither at $x = 0$.

\item
Show that $y = f(x)$ is a solution to the differential equation 
$$
xy' + y = \cos x.
$$

\end{enumerate}



\item
Let $f$ be the function given by 
$$
f(x) = \sin\left(  5x + \frac{\pi}{4} \right),
$$
and let $P(x)$ be the third-degree Taylor polynomial for $f$ centered about $x = 0$.
\begin{enumerate}

\item
Find $P(x)$.

\item 
Find the coefficient of $x^{22}$ in the Taylor series for $f$ about $x = 0$.

\item
Use the error bound to show that
$$
\left| f\left( \frac{1}{10} \right) - P\left(  \frac{1}{10} \right) \right| < \frac{1}{100}.
$$

\item
Let
$$
G(x) = \int_0^x f(t) \, dt.
$$
Write the third-degree Taylor polynomial for $G$ about $x = 0$.

\end{enumerate}

\item
If $f(x) = \cos x$ and $g(x) = \sin x$, find the first five terms of the power series for $f(x) g(x)$.


\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 10, 2006}



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