Taylor Series

Taylor Series

     The Taylor series for $f(x)$ about $x=a$ is defined as
$f(x)=f(a)+f'(a)(x-a)+\cfrac{f''(a)(x-a)^2}{2!}+\dots +\cfrac{f^{(n-1)}(a)(x-a)^{n-1}}{(n-1)!}+R_n$  (1)

where $R_n=\frac{f^{(n)}(x_0)(x-a)^n}{n!},$ $x_0$ between $a$ and $x$   (2)

is called the $remainder$ and where it is supposed that $f(x)$ has derivatives of order $n$ at least. The case where $n=1$ is often called the $law of the mean$ or $mean-value theorem$ and can be written as

$\cfrac{f(x)-f(a)}{x-a}=f'(x_0)$ $x_0$ between  $a$ and $x$

     The infinite series corresponding to $(1)$ , also called the $formal Taylor series$ for $f(x)$ will converge in some interval if $\displaystyle\lim_{n\to \infty}R_n =0$ in this interval. Some important Taylor series together with their intervals of convergence are as follows.

1. $e^x= 1+x+\cfrac{x^2}{2!}+\cfrac{x^3}{3!}+\cfrac{x^4}{4!}+\dots $	$-\infty < x< \infty $
2. $\sin x=x-\cfrac{x^3}{3!} +\cfrac{x^5}{5!} -\cfrac{x^7}{7!} +\dots $	$-\infty < x< \infty $
3. $\cos x=1-\cfrac{x^2}{2!}+\cfrac{x^4}{4!}-\cfrac{x^6}{6!}+\dots$	$-\infty < x< \infty $
4. $\ln (1+x)=x-\cfrac{x^2}{2}+\cfrac{x^3}{3}-\cfrac{x^4}{4}+\dots $	$-\infty < x< \infty $
5. $\tan ^{-1}x=x-\cfrac{x^3}{3}+\cfrac{x^5}{5}-\cfrac{x^7}{7}+\dots $	$-\infty < x< \infty $

A series of the from $\displaystyle\sum_{n=0}^{\infty}c_n(x-a)^n$ is often called a $power series$. Such power series are uniformly convergent  in any interval which lies entirely within the interval of convergence  

Functions Of Two Or More Variables

     The for example $z=f(x,y)$ defines a function $f$ which assigns to the number pair $(x,y)$ the number $z$ .

     Example If $f(x,y)=x^2+3xy+2y^2$ then $f(-1,2)=(-1)^2+3(-1)(2)+2(2)^2 =3$ 

     The student is familiar with graphing $z=f(x,y)$ in a 3-dimensional $xyz$ coordinate  system to obtain a $surface $. We sometimes call $x$ and $y$ $independent$ variable  and $z$ a symbol $z$ in two different sense . However , no confusion should result .

     The ideas of limits and continuity for functions of two or more variables pattern closely those for one variable.   

Partial

     The partial derivatives of $f(x,y)$ with respect to $x$ and $y$ are defined by

$\cfrac{\partial f}{\partial x}=\lim_{h\to \infty}\cfrac{f(x+h,y)-f(x,y)}{h},\cfrac{\partial f}{\partial y}=\lim_{k\to \infty}\cfrac{f(x,y+k)-f(x,y)}{k}$   $\dots$ (1)

     if these limits exist . We often write $h=\bigtriangleup x,k\bigtriangleup y$ . Note that $\partial f/\partial x$ is simple the ordinary derivative of $f$ with respect to $x$ keeping $y$ constant , while $\partial f/\partial y$ is the ordinary derivative of $f$ with respect to $y$ keeping $x$ constant.

     Example If $f(x,y)=3x^2-4xy+2y^2$ then $\cfrac{\partial f}{\partial x}=6x -4y,\cfrac{\partial f}{\partial y}=-4x+4y$

     Higher derivatives are defined similarly . For example , we have the second order derivatives

$\cfrac{\partial}{\partial x}\left(\cfrac{\partial f}{\partial x}\right)=\cfrac{\partial^2f}{\partial x^2},\cfrac{\partial }{\partial x}\left(\cfrac{\partial f}{\partial y}\right)=\cfrac{\partial^2f}{\partial x\partial y}$ ,$\cfrac{\partial}{\partial y}\left(\cfrac{\partial f}{\partial x}\right)=\cfrac{\partial^2f}{\partial y\partial x},\cfrac{\partial}{\partial y}\left(\cfrac{\partial f}{\partial y}\right)=\cfrac{\partial^2f}{\partial y^2}$   $\dots$(2)

     The derivatives in (1) are sometimes denoted by $f_x$ and $f_y$ . In such case $f_x(a,b),f_y(a,b)$ denote these partial derivatives evaluated at $(a,b)$. Similarly the derivatives in (2) are denoted by $f_{xx},f_{xy},f_{yx},f_{yy}$ respectively . The second and third results in (2) will be the same if $f$ has continuous partial derivatives of second order at least .

     The $differential $ of $f(x,y)$ is defined as

$df =\cfrac{\partial f}{\partial x}dx+\cfrac{\partial f}{\partial y}dy$   $\dots$ (3)

where $h=\bigtriangleup x=dx,k=\bigtriangleup y =dy$.

Generalizations of these results are easily made.

Taylor Series For Functions Of  Two Or More Variables 

     The ideas involved in Taylor series for functions of one variable can be generalize 

     For example ,the Taylor series for $f(x,y)$ about $x=a,y=b$ can be written 

$f(x,y)=f(a,b)+f_x(a,b)(x-a)+f_y(a,b)(y-b)+\cfrac{1}{2!}[f_{xx}(a,b)(x-a)^2+2f_{xy}(a,b)(x-a)(y-b)+f_{yy}(a,b)(y-b)^2]+\dots $ (4)