Theorem 1 (Taylor's Theorem: Bounding the Error)   Suppose $f$ and all its derivatives are continuous. If $T_n(x)$ is the $n$ -th Taylor polynomial of $f(x)$ around $x=a$ , then the error, or the difference between the real value of $f$ and the values of $T_n(x)$ is given by: $$|E_n(x)|=|f(x)-T_n(x)|\leq \frac{M}{(n+1)!}|x-a|^{n+1}$$ where $M$ is the maximum value of $f^{(n+1)}$ (the $n+1$ -th derivative of $f$ ) in the interval between $a$ and $x$ .

Example 2   Suppose we want to approximate $e$ using the Taylor polynomial of degree 4, $T_4(x)$ , around $x=0$ for the function $e^x$ . We know that $$T_4(x)=1+x+x^2/2+x^3/3!+x^4/4!$$ so we are asking how close are $e$ and $T_4(1)=1+1+1/2+1/6+1/24$ . In order to use the formula in the theorem, we just need to find $M$ , the maximum value of the $4$ th derivative of $e^x$ between $a=0$ and $x=1$ . Since $f^{(4)}=e^x$ and $e^x$ is strictly increasing, the maximum in $(0,1)$ happens at $x=1$ . Thus $M=e$ which is a number, say, less than $3$ . Therefore: $$|E_4(1)|=|f(1)-T_4(1)|=|e-(1+1+1/2+1/6+1/24)|\leq \frac{M}{5!}|1-0|^5=\frac{M}{5!}<\frac{3}{5!}=0.025$$ Thus the approximation has an error of less than $0.025$ . Actually, if we use a calculator we obtain that the error is exactly $0.0099$ . But, of course, the whole point of the theorem is not to use a calculator.

Example 3   What Taylor polynomial $T_n(x)$ (what $n$ ) should we use to approximate $e$ within $0.0001$ ? As above, we will be using the Taylor polynomial $T_n(x)$ for $e^x$ , evaluated at $x=1$ . Thus, we want the error $|E_n(1)|<0.0001$ . Notice all derivatives are $e^x$ and the maximum happens at $x=1$ , where $e^1=e$ , so for all derivatives $M<3$ . Hence by the theorem: $$|E_n(1)|=|f(1)-T_n(1)|=|e-(1+1+1/2+\ldots+1/n!)|\leq \frac{M}{(n+1)!}|1-0|^{n+1}=\frac{M}{(n+1)!}<\frac{3}{(n+1)!}$$ So we need $3/(n+1)! < 0.0001$ . Try several values of $n$ until that is satisfied: $$ 3/2=1.5,\ 3/3!=0.5,\ 3/4!=0.125,\ 3/5!=0.025,\ 3/6! =0.00416$$ $$ 3/7!=0.00059, \quad 3/8!=0.00007$$ Thus $n=7$ should work. So we just need $T_7(x)$ , or add $1+1+1/2+\ldots+1/7!$ .