example of Taylor polynomials for the exponential function

Example 1.

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

$T_1(x)$$=$$1+x$

$T_2(x)$$=$$1+x+{\displaystyle \frac{x^2}{2}}$

$T_3(x)$$=$$1+x+{\displaystyle \frac{x^2}{2}}+{\displaystyle \frac{x^3}{3!}}=1+x+{\displaystyle \frac{x^2}{2}}+{\displaystyle \frac{x^3}{6}}$

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

In fact:

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

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

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

$e$$=$$2.718281828459045\mathrm{\dots }$

$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\overline{6}$

$e\approx T_4(1)$$=$$1+1+1/2+1/6+1/24=65/24=2.708333\overline{3}$

$e\approx T_5(1)$$=$$1+1+1/2+1/6+1/24+1/120=163/60=2.71666\overline{6}$