example of Taylor polynomials for the exponential functionExampleOfTaylorPolynomialsForTheExponentialFunction2005-02-21 11:05:582005-04-13 16:51:55ExampleTaylor seriesapproximations of e% this is the default PlanetMath preamble. as your knowledge
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\newcommand{\Rats}{\mathbb{Q}}\begin{exa}
We construct the $n$th Taylor polynomial for $f(x)=e^x$ around $x=0$. As we know all derivatives of $e^x$ equal $e^x$ and also, $e^0=1$. Therefore, $f^{(n)}(0)=1$ for any $n$. Thus:
\begin{eqnarray*}
T_1(x) &=& 1+x\\
T_2(x) &=& 1+x + \frac{x^2}{2}\\
T_3(x) &=& 1+x + \frac{x^2}{2}+ \frac{x^3}{3!}=1+x + \frac{x^2}{2} +\frac{x^3}{6}\\
T_4(x) &=& 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}
\end{eqnarray*}
In fact:
$$T_n(x)=1+x + \frac{x^2}{2}+ \frac{x^3}{3!}+ \frac{x^4}{4!}+\ldots+\frac{x^n}{n!}$$
\begin{center}
\includegraphics[scale=0.7]{expon}
Comparison of $e^x$ with the Taylor pol. of deg. $1$ (green), $2$ (blue) and $3$ (pink).
\end{center}
Let us use several Taylor polynomials to find approximations of the number $e$:
\begin{eqnarray*}
e &=& 2.718281828459045\ldots\\
e\approx T_1(1) &=& 1+1=2\\
e \approx T_2(1) &=& 1+1+1/2=2.5 \\
e \approx T_3(1) &=& 1+1+1/2+1/6=8/3=2.666\bar{6} \\
e \approx T_4(1) &=& 1+1+1/2+1/6+1/24=65/24=2.708333\bar{3} \\
e \approx T_5(1) &=& 1+1+1/2+1/6+1/24+1/120=163/60=2.71666\bar{6} \\
\end{eqnarray*}
\end{exa}