PlanetMath: example of Taylor polynomials for the exponential function

Example 1 We construct the th Taylor polynomial for around . As we know all derivatives of equal and also, . Therefore, $f^{(n)}(0)=1$ for any . Thus:

$\displaystyle T_1(x)$	$\displaystyle =$	$\displaystyle 1+x$
$\displaystyle T_2(x)$	$\displaystyle =$	$\displaystyle 1+x + \frac{x^2}{2}$
$\displaystyle T_3(x)$	$\displaystyle =$	$\displaystyle 1+x + \frac{x^2}{2}+ \frac{x^3}{3!}=1+x + \frac{x^2}{2} +\frac{x^3}{6}$
$\displaystyle T_4(x)$	$\displaystyle =$	$\displaystyle 1+x + \frac{x^2}{2}+ \frac{x^3}{3!}+ \frac{x^4}{4!}=1+x + \frac{x^2}{2}+ \frac{x^3}{6}+ \frac{x^4}{24}$

In fact:

$\displaystyle T_n(x)=1+x + \frac{x^2}{2}+ \frac{x^3}{3!}+ \frac{x^4}{4!}+\ldots+\frac{x^n}{n!}$

$\includegraphics[scale=0.7]{expon}$

Comparison of with the Taylor pol. of deg. (green), (blue) and (pink).

Let us use several Taylor polynomials to find approximations of the number :

$\displaystyle e$	$\displaystyle =$	$\displaystyle 2.718281828459045\ldots$
$\displaystyle e\approx T_1(1)$	$\displaystyle =$	$\displaystyle 1+1=2$
$\displaystyle e \approx T_2(1)$	$\displaystyle =$	$\displaystyle 1+1+1/2=2.5$
$\displaystyle e \approx T_3(1)$	$\displaystyle =$	$\displaystyle 1+1+1/2+1/6=8/3=2.666\bar{6}$
$\displaystyle e \approx T_4(1)$	$\displaystyle =$	$\displaystyle 1+1+1/2+1/6+1/24=65/24=2.708333\bar{3}$
$\displaystyle e \approx T_5(1)$	$\displaystyle =$	$\displaystyle 1+1+1/2+1/6+1/24+1/120=163/60=2.71666\bar{6}$