Chapter 8.8, Problem 21E

To determine

The number of terms of the Maclaurin series for $e^x$ that should to estimate $e^{0.1}$ to within $0.00001$

Answer to Problem 21E

need $4$ terms of the Maclaurin series for $e^x$ that should to estimate $e^{0.1}$ to within 0.00001

Explanation of Solution

Given information:

Use Taylor’s Inequality to determine the number of terms of the Maclaurin series for $e^x$ that should to estimate $e^{0.1}$ to within $0.00001$

Formula Used:

  $T_n(x)\to f(x)$ ???? ??&#??????????;&#??????????;

And $T_n(x)$ can be used as an approximation to

  $f:f(x)=T_n(x)$

Taylor’s Inequality, which says that if

  $\left|f^{(n+1)}(x)\right|\le M$ , then

  $\left|R_n(x)\right|\le \frac{M}{(n+1)!}{\left|x-a\right|}^{n+1}$

Calculation:

  $f(x)=e^x$

The nth derivative will be

  $f^n(x)=e^x$

Taylor’s Inequality at $x=0.1$

For Maclaurin Series, $a=0$

Start at $n=2$ and move up by $1$ until the remainder is less than $0.00001$

  $\begin{array}{l}\left|f^{n+1}(0.1)\right|\le M\\ e^{0.1}\le M\\ \left|R_2(x)\right|\le \frac{e^{0.1}}{3!}{\left|.1-0\right|}^4\\ \left|R_2(x)\right|\le 0.000005\end{array}$

So, need $4$ terms