example of Taylor polynomials for the exponential function

Example 1.

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:

 $\displaystyle T_{1}(x)$ $\displaystyle=$ $\displaystyle 1+x$ $\displaystyle T_{2}(x)$ $\displaystyle=$ $\displaystyle 1+x+\frac{x^{2}}{2}$ $\displaystyle T_{3}(x)$ $\displaystyle=$ $\displaystyle 1+x+\frac{x^{2}}{2}+\frac{x^{3}}{3!}=1+x+\frac{x^{2}}{2}+\frac{x% ^{3}}{6}$ $\displaystyle T_{4}(x)$ $\displaystyle=$ $\displaystyle 1+x+\frac{x^{2}}{2}+\frac{x^{3}}{3!}+\frac{x^{4}}{4!}=1+x+\frac{% x^{2}}{2}+\frac{x^{3}}{6}+\frac{x^{4}}{24}$

In fact:

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

Comparison of $e^{x}$ with the Taylor pol. of deg. $1$ (green), $2$ (blue) and $3$ (pink).

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

 $\displaystyle e$ $\displaystyle=$ $\displaystyle 2.718281828459045\ldots$ $\displaystyle e\approx T_{1}(1)$ $\displaystyle=$ $\displaystyle 1+1=2$ $\displaystyle e\approx T_{2}(1)$ $\displaystyle=$ $\displaystyle 1+1+1/2=2.5$ $\displaystyle e\approx T_{3}(1)$ $\displaystyle=$ $\displaystyle 1+1+1/2+1/6=8/3=2.666\bar{6}$ $\displaystyle e\approx T_{4}(1)$ $\displaystyle=$ $\displaystyle 1+1+1/2+1/6+1/24=65/24=2.708333\bar{3}$ $\displaystyle e\approx T_{5}(1)$ $\displaystyle=$ $\displaystyle 1+1+1/2+1/6+1/24+1/120=163/60=2.71666\bar{6}$
Title example of Taylor polynomials for the exponential function ExampleOfTaylorPolynomialsForTheExponentialFunction 2013-03-22 15:04:09 2013-03-22 15:04:09 alozano (2414) alozano (2414) 6 alozano (2414) Example msc 41A58 LogarithmFunction NaturalLogBase EIsTranscendental ExponentialFunction