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%TCIDATA{<META NAME="Title" CONTENT="Reference/Mathematics/Taylor and Maclaurin Series">}
%TCIDATA{Created=Mon Aug 19 14:52:24 1996}
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\begin{document}


\section{Taylor and Maclaurin Series}

\begin{theorem}
If $f$ has a power series representation (expansion) at $a$, that is, if 
\[
f\left( x\right) =\sum\limits_{n=0}^{\infty }c_{n}\left( x-a\right)
^{n}\qquad \left| x-a\right| <R
\]
then its coefficients are given by the formula 
\[
c_{n}=\frac{f^{\left( n\right) }\left( a\right) }{n!}
\]
\end{theorem}

Substituting the formula for $c_{n}$ back into the series, you obtain the
following form for $f$, called the \textbf{Taylor series of the function }$f$%
\textbf{\ at }$a$\textbf{\ }( or \textbf{about }$a$ or \textbf{centered at }$%
a$): \label{taylor series} 
\[
f\left( x\right) =\sum\limits_{n=0}^{\infty }\frac{f^{\left( n\right)
}\left( a\right) }{n!}\left( x-a\right) ^{n}=f\left( a\right) +\frac{f\prime
\left( a\right) }{1!}\left( x-a\right) +\frac{f\prime \prime \left( a\right) 
}{2!}\left( x-a\right) ^{2}+\frac{f\prime \prime \prime \left( a\right) }{3!}%
\left( x-a\right) ^{3}+... 
\]
In the special case where $a=0$, you obtain the following series, called the 
\textbf{Maclaurin series}: \label{maclaurin series} 
\[
f\left( x\right) =\sum\limits_{n=0}^{\infty }\frac{f^{\left( n\right)
}\left( 0\right) }{n!}x^{n}=f\left( 0\right) +\frac{f\prime \left( 0\right) 
}{1!}x+\frac{f\prime \prime \left( 0\right) }{2!}x^{2}+\frac{f\prime \prime
\prime \left( 0\right) }{3!}x^{3}+... 
\]

Functions that can be represented as power series about $a$ are called 
\textbf{analytic at }$a$. \label{analytic functions}Analytic functions are
infinitely differentiable at $a$; that is, they have derivatives of all
orders at $a$. However, not all infinitely differentiable functions are
analytic.

The partial sums of a Taylor series are given by 
\[
T_{n}\left( x\right) =\sum\limits_{i=0}^{n}\frac{f^{\left( i\right) }\left(
a\right) }{i!}\left( x-a\right) ^{i}=f\left( a\right) +\frac{f\prime \left(
a\right) }{1!}\left( x-a\right) +\frac{f\prime \prime \left( a\right) }{2!}%
\left( x-a\right) ^{2}+...+\frac{f^{\left( n\right) }\left( a\right) }{n!}%
\left( x-a\right) ^{n} 
\]
$T_{n}$ is a polynomial of degree $n$ called the $n$\textbf{th-degree Taylor
polynomial of }$f$\textbf{\ at }$a$.\label{nth-degree taylor polynomial}

\begin{theorem}
If $f\left( x\right) =T_{n}\left( x\right) +R_{n}\left( x\right) $, where $%
T_{n}$ is the $n$th-degree Taylor polynomial of $f$ at $a$ and 
\[
\lim\limits_{n\rightarrow \infty }R_{n}\left( x\right) =0
\]
for $\left| x-a\right| <R$, then $f$ is equal to the sum of its Taylor
series on the interval $\left| x-a\right| <R$; that is, $f$ is analytic at $a
$.
\end{theorem}

\begin{theorem}
(Taylor's Formula) \label{taylor's formula}If $f$ has $n+1$ derivatives in
an interval $I$ that contains the number $a$, then for $x$ in $I$ there is a
number $z$ strictly between $x$ and $a$ such that the remainder term in the
Taylor series can be expressed as 
\[
R_{n}\left( x\right) =\frac{f^{\left( n+1\right) }\left( z\right) }{\left(
n+1\right) !}\left( x-a\right) ^{n+1}
\]

\begin{remark}[1]
For the special case $n=0$, if you substitute $x=b$ and $z=c$ in Taylor's
Formula, you get 
\[
f\left( b\right) =f\left( a\right) +f\prime \left( c\right) \left(
b-a\right) 
\]
This is the \hyperref{Mean Value Theorem}{}{}{cal04_02.tex#Mean value theorem%
}.
\end{remark}

\begin{remark}[2]
The above expression for $R_{n}\left( x\right) $ is known as \textbf{%
Lagrange's form of the remainder term}.
\end{remark}
\end{theorem}

In applying Taylor's formula, the following equation is often useful:

\[
\lim\limits_{n\rightarrow \infty }\frac{x^{n}}{n!}=0\qquad \text{for every
real number }x 
\]

\subsubsection{Important Maclaurin Series\label{important maclaurin series}}

\begin{center}
\begin{tabular}{|c|c|}
\hline
Maclaurin Series & Interval of Convergence \\ \hline
$\frac{1}{1-x}=\sum\limits_{n=0}^{\infty }x^{n}=1+x+x^{2}+x^{3}+...$ & $%
\left( -1,1\right) $ \\ 
$e^{x}=\sum\limits_{n=0}^{\infty }\frac{x^{n}}{n!}=1+\frac{x}{1!}+\frac{x^{2}%
}{2!}+\frac{x^{3}}{3!}+...$ & $\left( -\infty ,\infty \right) $ \\ 
$\sin x=\sum\limits_{n=0}^{\infty }\left( -1\right) ^{n}\frac{x^{2n+1}}{%
\left( 2n+1\right) !}=x-\frac{x^{3}}{3!}+\frac{x^{5}}{5!}-\frac{x^{7}}{7!}%
+...$ & $\left( -\infty ,\infty \right) $ \\ 
$\cos x=\sum\limits_{n=0}^{\infty }\left( -1\right) ^{n}\frac{x^{2n}}{\left(
2n\right) !}=1-\frac{x^{2}}{2!}+\frac{x^{4}}{4!}-\frac{x^{6}}{6!}+...$ & $%
\left( -\infty ,\infty \right) $ \\ 
$\tan ^{-1}x=\sum\limits_{n=0}^{\infty }\left( -1\right) ^{n}\frac{x^{2n+1}}{%
2n+1}=x-\frac{x^{3}}{3}+\frac{x^{5}}{5}-\frac{x^{7}}{7}+...$ & $\left[
-1,1\right] $ \\ \hline
\end{tabular}
\end{center}

\begin{remark}
Notice that if you let $x=1$ in $e^{x}=\sum\limits_{n=0}^{\infty }\frac{x^{n}%
}{n!}=1+\frac{x}{1!}+\frac{x^{2}}{2!}+\frac{x^{3}}{3!}+...$, you have an
expression for the number $e$ as a sum of an infinite series: 
\[
e=\sum\limits_{n=0}^{\infty }\frac{1}{n!}=1+\frac{1}{1!}+\frac{1}{2!}+\frac{1%
}{3!}+...
\]
\bigskip 
\end{remark}

\subsection{Related Scientific Notebook Topics}

{\small \hyperref{Maclaurin series}{}{}{../../Help/DM7-7.tex#Power series}}

{\small \hyperref{Taylor series}{}{}{../../Help/DM7-7.tex#Taylor series}}

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